Quantum mechanics is the branch of science that deals with the behavior of matter and light on the atomic and subatomic scales. It attempts to describe and account for the properties of molecules and atoms and their constituents—electrons, protons, neutrons, and other more esoteric particles such as quarks and gluons. These properties include the interactions of the particles with one another and with electromagnetic radiation (i.e., light, X-rays, and gamma rays).
Table of Contents
The Peculiar Behavior of Matter and Radiation
The behavior of matter and radiation on the atomic scale often seems peculiar, and the consequences of quantum theory are accordingly difficult to understand and to believe. Its concepts frequently conflict with common-sense notions derived from observations of the everyday world. There is no reason, however, why the behavior of the atomic world should conform to that of the familiar, large-scale world. It is important to realize that quantum mechanics is a branch of physics and that the business of physics is to describe and account for the way the world—on both the large and the small scale—is and not how one imagines it or would like it to be.
The Rewards of Studying Quantum Mechanics
The study of quantum mechanics is rewarding for several reasons. First, it illustrates the essential methodology of physics. Second, it has been enormously successful in giving correct results in practically every situation to which it has been applied. There is, however, an intriguing paradox. Despite the overwhelming practical success of quantum mechanics, the foundations of the subject contain unresolved problems—in particular, problems concerning the nature of measurement.
An essential feature of quantum mechanics is that it is generally impossible, even in principle, to measure a system without disturbing it; the detailed nature of this disturbance and the exact point at which it occurs are obscure and controversial. Thus, quantum mechanics attracted some of the ablest scientists of the 20th century, and they erected what is perhaps the finest intellectual edifice of the period.
Historical Basis of Quantum Theory
Basic Considerations
At a fundamental level, both radiation and matter have the characteristics of particles and waves. The gradual recognition by scientists that radiation has particle-like properties and that matter has wavelike properties provided the impetus for the development of quantum mechanics. Influenced by Newton, most physicists of the 18th century believed that light consisted of particles, which they called corpuscles. From about 1800, evidence began to accumulate for a wave theory of light.
The Wave Theory of Light
At about this time, Thomas Young showed that, if monochromatic light passes through a pair of slits, the two emerging beams interfere, so that a fringe pattern of alternately bright and dark bands appears on a screen. The bands are readily explained by a wave theory of light. According to the theory, a bright band is produced when the crests (and troughs) of the waves from the two slits arrive together at the screen; a dark band is produced when the crest of one wave arrives at the same time as the trough of the other, and the effects of the two light beams cancel.
Beginning in 1815, a series of experiments by Augustin-Jean Fresnel of France and others showed that, when a parallel beam of light passes through a single slit, the emerging beam is no longer parallel but starts to diverge; this phenomenon is known as diffraction. Given the wavelength of the light and the geometry of the apparatus (i.e., the separation and widths of the slits and the distance from the slits to the screen), one can use the wave theory to calculate the expected pattern in each case; the theory agrees precisely with the experimental data.
Early Developments in Quantum Mechanics
Planck’s Radiation Law
By the end of the 19th century, physicists almost universally accepted the wave theory of light. However, though the ideas of classical physics explain interference and diffraction phenomena relating to the propagation of light, they do not account for the absorption and emission of light. All bodies radiate electromagnetic energy as heat; in fact, a body emits radiation at all wavelengths.
The energy radiated at different wavelengths is a maximum at a wavelength that depends on the temperature of the body; the hotter the body, the shorter the wavelength for maximum radiation. Attempts to calculate the energy distribution for the radiation from a blackbody using classical ideas were unsuccessful. (A blackbody is a hypothetical ideal body or surface that absorbs and reemits all radiant energy falling on it.) One formula, proposed by Wilhelm Wien of Germany, did not agree with observations at long wavelengths, and another, proposed by Lord Rayleigh (John William Strutt) of England, disagreed with those at short wavelengths.
The Birth of Quantum Theory
In 1900, the German theoretical physicist Max Planck made a bold suggestion. He assumed that the radiation energy is emitted, not continuously, but rather in discrete packets called quanta. The energy (E) of the quantum is related to the frequency $$(\nu)$$ by the equation:
[ $$E = h\nu$$ ]
The quantity (h), now known as Planck’s constant, is a universal constant with the approximate value of ($$6.62607 \times 10^{-34}$$) joule∙second. Planck showed that the calculated energy spectrum then agreed with observation over the entire wavelength range.
The development of quantum mechanics has been a journey from the macroscopic theories of Newton and the wave theory of light to the revolutionary ideas of Planck. This field has proven essential in understanding the fundamental properties of matter and radiation, providing a framework that explains the peculiar behaviors observed at atomic and subatomic scales. Despite its complexity and the unresolved questions it raises, quantum mechanics stands as a testament to the power of human ingenuity in uncovering the mysteries of the universe.
Einstein and the Photoelectric Effect
In 1905, Albert Einstein extended Max Planck‘s hypothesis to explain the photoelectric effect, which is the emission of electrons by a metal surface when it is irradiated by light or more energetic photons. The kinetic energy of the emitted electrons depends on the frequency ($$\nu$$) of the radiation, not on its intensity; for a given metal, there is a threshold frequency ($$\nu_0$$) below which no electrons are emitted. Furthermore, emission takes place as soon as the light shines on the surface; there is no detectable delay.
Einstein showed that these results can be explained by two assumptions:
- Light is composed of corpuscles or photons, the energy of which is given by Planck’s relationship:
[ $$E = h\nu$$ ] - An atom in the metal can absorb either a whole photon or nothing. Part of the energy of the absorbed photon frees an electron, which requires fixed energy ($$W$$), known as the work function of the metal; the rest is converted into the kinetic energy ($$\frac{1}{2} m_e u^2$$) of the emitted electron (where ($$m_e$$) is the mass of the electron and ($$u$$) is its velocity). Thus, the energy relation is:
[ $$h\nu = W + \frac{1}{2} m_e u^2$$ ]
If ($$\nu$$) is less than ($$\nu_0$$), where ($$h\nu_0 = W$$), no electrons are emitted. Not all the experimental results mentioned above were known in 1905, but all of Einstein’s predictions have been verified since.
Bohr’s Theory of the Atom
A major contribution to the subject was made by Niels Bohr of Denmark, who applied the quantum hypothesis to atomic spectra in 1913. The spectra of light emitted by gaseous atoms have been studied extensively since the mid-19th century. It was found that radiation from gaseous atoms at low pressure consists of a set of discrete wavelengths, known as a line spectrum because the radiation (light) emitted consists of a series of sharp lines. The wavelengths of the lines are characteristic of the element and may form extremely complex patterns. The simplest spectra are those of atomic hydrogen and the alkali atoms (e.g., lithium, sodium, and potassium).
For hydrogen, the wavelengths ($$\lambda$$) are given by the empirical formula:
[ $$\frac{1}{\lambda} = R_{\infty} \left( \frac{1}{m^2} – \frac{1}{n^2} \right)$$]
where ($$m$$) and ($$n$$) are positive integers with ($$n > m$$) and ($$R_{\infty}$$), known as the Rydberg constant, has the value ($$1.097373157 \times 10^7 \, \text{m}^{-1}$$). For a given value of ($$m$$), the lines for varying ($$n$$) form a series. The lines for ($$m = 1$$), the Lyman series, lie in the ultraviolet part of the spectrum; those for ($$m = 2$$), the Balmer series, lie in the visible spectrum; and those for ($$m = 3$$), the Paschen series, lies in the infrared.
Bohr started with a model suggested by the New Zealand-born British physicist Ernest Rutherford. The model was based on the experiments of Hans Geiger and Ernest Marsden, who in 1909 bombarded gold atoms with massive, fast-moving alpha particles; when some of these particles were deflected backward, Rutherford concluded that the atom had a massive, charged nucleus. In Rutherford’s model, the atom resembles a miniature solar system with the nucleus acting as the Sun and the electrons as the circulating planets.
Bohr’s Postulates
Bohr made three assumptions:
- Stationary States: In contrast to classical mechanics, where an infinite number of orbits is possible, an electron can be in only one of a discrete set of orbits, which he termed stationary states.
- Quantized Angular Momentum: The only orbits allowed are those for which the angular momentum of the electron is a whole number (n) times (\hbar) ((\hbar = \frac{h}{2\pi})).
- Classical Mechanics: Newton’s laws of motion, so successful in calculating the paths of the planets around the Sun, also applied to electrons orbiting the nucleus. The force on the electron (the analog of the gravitational force between the Sun and a planet) is the electrostatic attraction between the positively charged nucleus and the negatively charged electron.
With these simple assumptions, he showed that the energy of the orbit has the form:
[ $$E_n = -\frac{E_0}{n^2}$$ ]
where ($$E_0$$) is a constant that may be expressed by a combination of the known constants ($$e$$), ($$m_e$$), and ($$\hbar$$).
Radiation and Energy Transitions
While in a stationary state, the atom does not give off energy as light; however, when an electron makes a transition from a state with energy ($$E_n$$) to one with lower energy ($$E_m$$), a quantum of energy is radiated with frequency ($$\nu$$), given by the equation:
[$$ h\nu = E_n – E_m $$]
Inserting the expression for ($$E_n$$) into this equation and using the relation ($$\lambda \nu = c$$), where (c) is the speed of light, Bohr derived the formula for the wavelengths of the lines in the hydrogen spectrum, with the correct value of the Rydberg constant.
Impact of Bohr’s Theory
Bohr’s theory was a brilliant step forward. Its two most important features have survived in present-day quantum mechanics:
- Stationary States: The existence of stationary, non-radiating states.
- Radiation Frequency: The relationship of radiation frequency to the energy difference between the initial and final states in a transition. Before Bohr, physicists had thought that the radiation frequency would be the same as the electron’s frequency of rotation in an orbit.
Scattering of X-rays
In 1912, Max von Laue of Germany demonstrated that crystals can act as three-dimensional diffraction gratings for X-rays, thereby providing fundamental evidence for their wavelike nature. In his experiment, the regular lattice arrangement of atoms in a crystal scatters X-rays, leading to constructive interference when the X-ray crests coincide. This results in intense scattered X-ray beams in certain directions, clearly demonstrating wave behavior. By knowing the interatomic distances in the crystal and the directions of constructive interference, the wavelength of the X-rays can be calculated.
In 1922, Arthur Holly Compton of the United States provided evidence for the particle nature of X-rays. He conducted experiments on the scattering of monochromatic, high-energy X-rays by graphite and observed that part of the scattered radiation retained the same wavelength ($$\lambda_0$$) as the incident X-rays, while another component had a longer wavelength ($$\lambda$$). Compton interpreted this by treating the X-ray photon as a particle that collides with an electron in the graphite, akin to billiard balls. By applying the laws of conservation of energy and momentum, he derived the relationship between the energy transferred to the electron and the scattering angle ($$\theta$$):
$$[ \lambda’ – \lambda_0 = \frac{h}{m_ec} (1 – \cos \theta) ]$$
where ($$h$$) is Planck’s constant, ($$m_e$$) is the electron mass, and ($$c$$) is the speed of light. This experimental confirmation of Compton’s formula is direct evidence of the corpuscular nature of radiation.
De Broglie’s Wave Hypothesis
In 1924, faced with the dual particle and wave nature of electromagnetic radiation, Louis-Victor de Broglie of France proposed a unifying hypothesis: matter also has both particle and wave properties. He suggested that material particles can behave as waves, with their wavelength ($$\lambda$$) related to their linear momentum ($$p$$) by the equation:
[ $$\lambda = \frac{h}{p}$$]
In 1927, Clinton Davisson and Lester Germer of the United States confirmed de Broglie’s hypothesis for electrons. They diffracted a beam of monoenergetic electrons using a nickel crystal and showed that the wavelength of the waves is related to the momentum of the electrons by de Broglie’s equation. Since their investigation, similar experiments have been performed with atoms, molecules, neutrons, protons, and many other particles, all behaving like waves with the same wavelength-momentum relationship.
Basic Concepts and Methods in Quantum Mechanics
Niels Bohr‘s theory, which initially assumed that electrons moved in circular orbits, was extended by the German physicist Arnold Sommerfeld and others to include elliptic orbits and other refinements. Attempts were made to apply this theory to more complex systems than the hydrogen atom. However, the ad hoc mixture of classical and quantum ideas made the theory and calculations increasingly unsatisfactory.
Then, during a remarkably creative period beginning in July 1925, a series of papers by German scientists set the subject on a firm conceptual foundation. Two distinct approaches emerged: matrix mechanics, proposed by Werner Heisenberg, Max Born, and Pascual Jordan, and wave mechanics, put forward by Erwin Schrödinger. The proponents of these theories were often critical of each other’s work. Heisenberg found Schrödinger’s ideas “disgusting,” while Schrödinger was “discouraged and repelled” by the lack of visualization in Heisenberg’s method. Despite these differences, Schrödinger demonstrated that the two theories are mathematically equivalent.
Schrödinger’s Wave Mechanics
The present discussion follows Schrödinger’s wave mechanics, as it is less abstract and easier to understand than Heisenberg’s matrix mechanics. Schrödinger’s wave mechanics provides a comprehensive framework for understanding the behavior of particles at the quantum level and remains a cornerstone of modern quantum theory.
Formulation of Schrödinger’s Equation
Erwin Schrödinger translated de Broglie‘s hypothesis about the wave nature of matter into a mathematical framework that could address a variety of physical problems without additional arbitrary assumptions. Guided by the analogy of optics, where the propagation of light rays is derived from wave motion when the wavelength is small relative to the dimensions of the apparatus, Schrödinger sought a wave equation for a matter that would reduce to particle-like behavior when the wavelength is comparatively small.
In classical mechanics, if a particle of mass ( $$m_e$$ ) is subjected to a force such that its potential energy is $$V(x, y, z)$$ at position $$(x, y, z)$$, then the sum of $$V(x, y, z)$$ and the kinetic energy ($$\frac{p^2}{2m_e}$$) is equal to a constant, the total energy ( $$E$$ ) of the particle:
[ $$E = V(x, y, z) + \frac{p^2}{2m_e}$$ ]
Assuming the particle is bound, meaning it is confined within a region due to insufficient energy to escape, Schrödinger replaced the momentum ( $$p$$ ) in this equation with a differential operator, incorporating de Broglie’s relation. He formulated the wave function ($$\Psi(x, y, z)$$) which satisfies the time-independent Schrödinger equation:
[$$ -\frac{\hbar^2}{2m_e} \nabla^2 \Psi + V \Psi = E \Psi $$]
where ($$\hbar$$) is the reduced Planck constant and ($$\nabla^2$$) is the Laplacian operator. This equation laid the foundation for quantum mechanics in a broadly applicable manner.
Application to the Hydrogen Atom
Schrödinger applied his equation to the hydrogen atom, where the potential function is given by:
[ $$V(r) = -\frac{e^2}{r}$$ ]
Here, ( $$-e$$ ) represents the charge of the electron, ( $$e$$ ) is the charge of the nucleus (proton), and ( $$r$$ ) is the distance from the nucleus. Solving the Schrödinger equation for this potential with non-trivial mathematics, Schrödinger found that only certain discrete values of ( $$E$$ ) lead to acceptable wave functions ($$\Psi$$). These functions are characterized by three quantum numbers ( n ), ( l ), and ( m ):
- ( n ): Principal quantum number (1, 2, 3, …)
- ( l ): Orbital angular momentum quantum number
- ( m ): Magnetic quantum number
The energy levels ( $$E_n$$ ) depend solely on ( n ) and match those predicted by Bohr’s theory. The angular momentum is given by ($$\sqrt{l(l + 1)} \hbar$$), and the component of angular momentum along a chosen axis is ( $$m \hbar$$ ).
Interpretation of the Wave Function
The square of the wave function, ($$\Psi^2$$), provides a physical interpretation. Max Born proposed that ($$\Psi^2$$) represents the probability density of finding an electron at a position $$(x, y, z)$$, rather than implying a smeared-out density of the electron. This interpretation replaces the idea of the electron as a point particle with the concept of electron “clouds” that represent probable locations.
Electron Spin and Antiparticles
In 1928, Paul A.M. Dirac formulated a wave equation for the electron that incorporated relativity and quantum mechanics. Schrödinger’s equation, based on a nonrelativistic expression for kinetic energy $$(\frac{p^2}{2m_e})$$, does not satisfy the requirements of special relativity. Dirac introduced an additional quantum number ( $$m_s$$ ) for electron spin, which can take values of ( $$+\frac{1}{2}$$ ) or ( $$-\frac{1}{2}$$ ). This spin angular momentum was initially introduced by Samuel A. Goudsmit and George E. Uhlenbeck to explain magnetic moment measurements that could not be accounted for by orbital angular momentum alone.
The Dirac equation also predicted the existence of antiparticles. In 1932, Carl David Anderson discovered the positron, the antiparticle of the electron, which has the same mass as the electron but opposite charge.
Identical Particles and Multielectron Atoms
In quantum mechanics, identical particles, such as electrons, are indistinguishable from each other. The wave function ($$\Psi$$) for a system of identical particles must satisfy special symmetry conditions. Interchanging the coordinates of two identical particles should either leave the wave function unchanged (symmetric) or change its sign (antisymmetric).
- Fermions (e.g., electrons, protons, neutrons) have antisymmetric wave functions and obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state simultaneously. This principle is fundamental to many properties of matter, including the periodic table and the behavior of solids.
- Bosons (e.g., photons, mesons) have symmetric wave functions and do not obey the exclusion principle.
The Schrödinger equation is challenging to solve precisely for atoms with more than one electron due to the complexity of interactions. Approximation methods, such as those developed by Douglas R. Hartree and Vladimir Fock, have been successful. These methods involve assuming each electron moves independently in an average field created by the nucleus and other electrons. The total wave function must comply with the exclusion principle, and corrections are applied to account for electron-electron interactions and magnetic forces.
Time-Dependent Schrödinger Equation
In 1926, Erwin Schrödinger introduced a general wave equation to describe quantum systems that evolve. This equation extends the earlier time-independent Schrödinger equation to account for the time-dependent behavior of quantum systems.
The time-dependent Schrödinger equation is expressed as:
$$
i \hbar \frac{\partial \Psi(x, y, z, t)}{\partial t} = \hat{H} \Psi(x, y, z, t)
$$
where:
- ($$i$$) is the imaginary unit $$(\sqrt{-1})$$,
- $$(\hbar)$$ is the reduced Planck constant $$(\hbar = \frac{h}{2\pi})$$,
- $$\Psi(x, y, z, t)$$ is the wave function of the particle as a function of position $$(x, y, z)$$ and time $$(t)$$,
- $$(\hat{H})$$ is the Hamiltonian operator, representing the total energy of the system.
Time-Independent Solution
For systems with constant energy $$(E)$$, the wave function can be separated into spatial and temporal parts. The solution to the time-dependent Schrödinger equation takes the form:
$$
\Psi(x, y, z, t) = \psi(x, y, z) \exp\left(-\frac{i E t}{\hbar}\right)
$$
where exp stands for the exponential function, and the time-dependent Schrödinger equation reduces to the time-independent form.
Substituting this into the time-dependent Schrödinger equation, we obtain the time-independent Schrödinger equation:
$$
\hat{H} \psi(x, y, z) = E \psi(x, y, z)
$$
Transition Probabilities
The time-dependent Schrödinger equation is useful for calculating the probability of transitions between different stationary states of an atom. When an atom transitions from a higher energy state to a lower one, it emits a photon with energy corresponding to the difference between these states. Conversely, if electromagnetic radiation of the right frequency is applied, it can stimulate transitions, leading to absorption or emission of photons. These processes form the basis for technologies such as lasers.
Selection Rules
The probability of transitions between states depends on the quantum numbers associated with the initial and final states. Selection rules dictate the allowed transitions, which are based on the conservation of angular momentum. For instance, one key selection rule states that the orbital angular momentum quantum number $$(l)$$ must change by one unit $$(\Delta l = \pm 1)$$, reflecting the fact that photons have spin 1.
Quantum Tunneling
Quantum tunneling is a phenomenon with no classical counterpart. Consider a particle in a one-dimensional potential well $$V(x)$$, as illustrated below:
- Classical Mechanics: If the particle’s energy $$(E)$$ is less than the potential barrier height $$(V_0)$$, the particle remains confined within the well. If $$(E > V_0)$$, it can escape.
- Quantum Mechanics: Even if $$(E < V_0)$$, the particle has a nonzero probability of escaping through the barrier. This effect, known as tunneling, occurs because the particle’s wave function has a finite probability density beyond the barrier, allowing it to tunnel through and emerge with the same energy $$(E)$$.
Tunneling and Its Applications
Tunneling Phenomenon
The phenomenon of tunneling has significant implications in quantum mechanics and various practical applications. One notable example is radioactive decay, specifically alpha decay. In this process, a nucleus emits an alpha particle (a helium nucleus), and tunneling provides a quantum mechanical explanation for this event.
George Gamow and Ronald W. Gurney with Edward Condon independently developed a quantum mechanical model in 1928 to describe this phenomenon. According to their model, the alpha particle is initially confined within a potential barrier, as illustrated below.
Tunneling and Alpha Decay
For a given nuclear species, the energy $$E$$ of the emitted alpha particle and the average lifetime $$\tau$$ of the nucleus before decay can be measured. The lifetime $$\tau$$ is an indicator of the probability of tunneling through the potential barrier. The shorter the lifetime, the higher the tunneling probability.
The relationship between $$\tau$$ and $$E$$ can be derived based on the potential function, and experimental data supports this theory. The energy $$E$$ of emitted alpha particles typically ranges from 2 to 8 million electron volts (MeV), where 1 MeV equals $$10^6$$ electron volts. Despite the relatively small variation in $$E$$, ranging by a factor of 4, $$\tau$$ varies enormously—from approximately $$10^{11}$$ years to $$10^{-6}$$ seconds, a factor of $$10^{24}$$. This extraordinary sensitivity is effectively explained only by quantum mechanical tunneling.
Axiomatic Approach to Quantum Mechanics
Dirac’s Axiomatic Framework
Paul A.M. Dirac provided a profound and general formulation of quantum mechanics through an axiomatic approach in his classic textbook, The Principles of Quantum Mechanics (1930). This framework is based on observables and states.
Observables and States
An observable is any measurable quantity, such as energy, position, or angular momentum. Each observable is associated with a set of states, each represented by an algebraic function. For an observable with $$N$$ states, denoted by $$\psi_1, \psi_2, \ldots, \psi_N$$, and corresponding measurement values $$a_1, a_2, \ldots, a_N$$, the physical system can be represented by a wave function $$\Psi$$ that is a linear combination of these states:
$$
\Psi = c_1 \psi_1 + c_2 \psi_2 + \cdots + c_N \psi_N
$$
where $$c_1, c_2, \ldots, c_N$$ are coefficients that can be calculated. Although these coefficients are generally complex, they are assumed to be real for simplicity in this discussion.
Measurement and Probability
According to Dirac’s postulates:
- The result of a measurement must be one of the eigenvalues $$a_i$$ of the observable.
- Before measurement, the probability of obtaining a particular value $$a_i$$ is $$|c_i|^2$$.
After measuring a value, say $$a_5$$, the system’s state changes to $$\psi_5$$. This change implies that the measurement process alters the state of the system in an indeterminate manner, contradicting classical physics’ expectation that measurements yield a definite result.
Special Case: Definite States
An exception arises when the system’s state $$\Psi$$ is already one of the eigenstates, say $$\psi_3$$. In this case, $$c_3 = 1$$ and all other $$c_i$$ are zero. Here, the measurement of the observable yields the value $$a_3$$ with certainty, and the system remains in the state $$\psi_3$$ after the measurement. Thus, in this specific scenario, measurement does not disturb the system, and consecutive measurements produce consistent results.
Measurement of Multiple Observables
If two observables share the same set of eigenstates $$\psi_1, \psi_2, \ldots, \psi_N$$, measuring one observable yields a state that is also an eigenstate of the second observable. Thus, both observables’ values can be determined simultaneously with certainty. The eigenvalues for the two observables may differ, but their simultaneous measurement does not disturb the system after the first measurement.
Incompatible Observables
Measurement of Incompatible Observables
When measuring two observables that have different sets of state functions, the situation becomes complex. Measuring one observable gives a definite result, but this result alters the system’s state, making it unsuitable for measuring the second observable. Consequently, the two observables are termed incompatible, meaning that both quantities cannot be known simultaneously.
Example: Angular Momentum Components
A specific example of incompatible observables is the measurement of angular momentum components along mutually perpendicular directions. The Stern-Gerlach experiment provides insight into this phenomenon. This experiment involves measuring the spin angular momentum of silver atoms.
Stern-Gerlach Experiment
In this setup, a beam of silver atoms is passed through a magnetic field with varying strength over a small distance (Figure 2). The experiment determines the spin quantum number $$m_s$$, which can take values of $$+1/2$$ or $$−1/2$$. The magnetic field produces a force on the silver atoms depending on their spin state, resulting in two distinct beams (Figure 3):
- Beam 1: Atoms with $$m_s = +1/2$$ are deflected upward.
- Beam 2: Atoms with $$m_s = −1/2$$ are deflected downward.
If the magnetic field is aligned along the $$x$$-axis, the apparatus measures the $$x$$-component of spin angular momentum $$S_x$$. For these atoms:
$$
S_x = \pm \frac{\hbar}{2}
$$
where $$\hbar$$ is the reduced Planck constant. In classical terms, these states correspond to atoms spinning around the $$x$$-axis with opposite rotational senses.
Measurement Along the y-axis
Similarly, the $$y$$-component of spin angular momentum $$S_y$$ can also have values of $$+ \frac{\hbar}{2}$$ and $$− \frac{\hbar}{2}$$. However, the states for $$S_y$$ are not the same as for $$S_x$$. Each $$S_x$$ state is an equal mixture of the $$S_y$$ states, and vice versa. This mixing implies that classical pictures of quantum states do not fully capture the quantum reality.
Quantum Measurement and Disturbance
Consider a scenario where atoms from Beam 1 are passed through a second magnet (Magnet B) with a magnetic field along the $$y$$-axis. According to classical theory, this would measure both the $$x$$- and $$y$$-components of spin angular momentum simultaneously, producing:
- Beam 3: Atoms with $$S_x = + \frac{\hbar}{2}$$ and $$S_y = + \frac{\hbar}{2}$$.
- Beam 4: Atoms with $$S_x = + \frac{\hbar}{2}$$ and $$S_y = – \frac{\hbar}{2}$$.
However, classical theory fails to account for the quantum mechanical principle that measurement disturbs the system. If Beam 3 is subsequently passed through another magnet (Magnet C) aligned along the $$x$$-axis, the atoms split equally into:
- Beam 5: Atoms with $$S_x = + \frac{\hbar}{2}$$.
- Beam 6: Atoms with $$S_x = – \frac{\hbar}{2}$$.
This indicates that the beam entering Magnet B, although known to have $$S_x = + \frac{\hbar}{2}$$, is actually an equal mixture of states with $$S_y = + \frac{\hbar}{2}$$ and $$S_y = – \frac{\hbar}{2}$$. Conversely, after exiting Magnet B, the beam has $$S_y = + \frac{\hbar}{2}$$ but is an equal mixture of states with $$S_x = + \frac{\hbar}{2}$$ and $$S_x = – \frac{\hbar}{2}$$.
Thus, the measurement of $$S_y$$ in Magnet B disturbs the previously known value of $$S_x$$. The key point is that knowing one component of spin angular momentum leads to uncertainty in the measurement of the perpendicular component.
Heisenberg Uncertainty Principle
Discrete and Continuous Observables
In quantum mechanics, observables such as the energy of a bound system and angular momentum components have discrete sets of experimental values. For instance, the energy of a bound system is discrete, and angular momentum components take the form $$m\hbar$$, where $$m$$ is an integer or half-integer, positive or negative. However, the position and linear momentum of a free particle can take continuous values in both quantum and classical theory.
The mathematics of observables with a continuous spectrum of measured values is more complex but principled. An observable with a continuous spectrum has an infinite number of state functions. The state function $$\Psi$$ of the system is considered a combination of the state functions of the observable, but the sum must be replaced by an integral.
Measurement of Position and Momentum
Measurements of a particle’s position $$x$$ and the $$x$$-component of its linear momentum $$p_x$$ are incompatible because they have different state functions. This incompatibility is illustrated by the phenomenon of diffraction. When a parallel monochromatic light beam passes through a slit (Figure 4A), its intensity varies with direction, showing zero intensity in certain directions (Figure 4B). The first zero occurs at an angle $$\theta_0$$, given by:
$$
\sin \theta_0 = \frac{\lambda}{b}
$$
where $$\lambda$$ is the wavelength of the light, and $$b$$ is the width of the slit. If the slit width is reduced, $$\theta_0$$ increases, indicating a more spread-out diffracted light beam.
This experiment can be replicated with a stream of electrons, which exhibit wave-like properties according to de Broglie. The electrons’ beam spreads out like light waves, and their momentum is $$p = me u$$, where $$me$$ is the electron’s mass and $$u$$ is its velocity in the forward direction (the $$y$$-direction). For the $$x$$-component of momentum $$p_x$$, the uncertainty after passing through the slit is:
$$
\Delta p_x \approx \frac{h}{b}
$$
where $$h$$ is Planck’s constant. The exact position where an electron passed through the slit is unknown, so the uncertainty in $$x$$-position is:
$$
\Delta x \approx \frac{b}{2}
$$
Thus, the product of uncertainties is of the order of $$\hbar$$, leading to the Heisenberg Uncertainty Principle:
$$
\Delta x \Delta p_x \geq \frac{\hbar}{2}
$$
This principle states a fundamental limit to the precision with which position and momentum can be measured simultaneously. Increasing the precision of one measurement decreases the precision of the other.
Significance of the Uncertainty Principle
The uncertainty principle is crucial on the atomic scale due to the small value of $$h$$ in everyday units. For a macroscopic object with a mass of one gram, if its position is measured with a precision of $$10^{-6}$$ meters, the uncertainty in its velocity is about $$10^{-25}$$ meters per second, which is negligible. However, for an electron in an atom about $$10^{-10}$$ meters across, the minimum uncertainty in velocity is about $$10^6$$ meters per second.
Wave-Particle Duality and Measurement
The uncertainty principle is rooted in the wave-particle duality of particles like electrons and photons. When Heisenberg first proposed the principle in 1927, he considered measuring an electron’s position using a microscope. Due to diffraction, the image is blurred, and the uncertainty in position is approximately the light’s wavelength. Using shorter wavelengths (like gamma rays) reduces this uncertainty, but the resulting Compton effect increases momentum uncertainty.
A detailed calculation yields the same uncertainty relation:
$$
\Delta E \Delta t \geq \frac{\hbar}{2}
$$
where $$\Delta E$$ is the energy uncertainty and $$\tau$$ is the measurement duration.
Energy Spread and Natural Broadening
The Schrödinger and Dirac theories provide precise energy values for stationary states, but in reality, these states have an energy spread due to finite lifetimes. The energy spread $$\Delta E$$ is related to the mean lifetime $$\tau$$ of the state by:
$$
\Delta E \tau \geq \frac{\hbar}{2}
$$
This relation explains the natural broadening of spectral lines, which is fundamental and cannot be reduced. The uncertainty principle highlights the intrinsic limitations of measurements at quantum scales.
Quantum Electrodynamics
Introduction to QED
Quantum Electrodynamics (QED) applies quantum theory to the interaction between electrons and radiation. This requires a quantum treatment of Maxwell’s field equations, foundational to electromagnetism, and Dirac’s relativistic theory of the electron. QED is a quantum field theory that explains the behavior and interactions of electrons, positrons, and photons. It addresses processes involving the creation of material particles from electromagnetic energy and the annihilation of a material particle with its antiparticle, producing energy.
Challenges and Renormalization
Initially, QED faced significant mathematical challenges, as calculated values for quantities such as the electron’s charge and mass were infinite. An innovative set of techniques, developed in the late 1940s by Hans Bethe, Julian S. Schwinger, Tomonaga Shin’ichirō, Richard P. Feynman, and others, systematically handled these infinities to obtain finite physical quantities. This method, known as renormalization, has led to remarkably accurate predictions.
Energy States and the Lamb Shift
According to Dirac’s theory, two specific states in hydrogen with different quantum numbers have the same energy. However, QED predicts a slight difference in their energies, measurable by the frequency of the electromagnetic radiation causing transitions between the two states. Willis E. Lamb, Jr., and Robert Retherford first measured this effect, known as the Lamb shift, in 1947.
The physical origin of the Lamb shift lies in the interaction of the electron with random fluctuations in the surrounding electromagnetic field. These fluctuations are a quantum phenomenon that exists even in the absence of an applied field. The accuracy of both experiment and theory in this area is exemplified by recent values for the separation of the two states, expressed in terms of the transition frequency:
$$ \Delta E = h \nu $$
Magnetic Dipole Moment of the Electron
A more striking example of QED’s success is the value for $$\mu_e$$, the magnetic dipole moment of the free electron. Because the electron spins and has an electric charge, it acts like a tiny magnet, with its strength expressed by $$\mu_e$$. According to Dirac’s theory, $$\mu_e$$ is exactly equal to:
$$ \mu_B = \frac{e\hbar}{2m_e} $$
This quantity is known as the Bohr magneton. However, QED predicts:
$$ \mu_e = (1 + a) \mu_B $$
where $$a$$ is a small number, approximately $$\frac{1}{860}$$. The QED correction originates from the interaction of the electron with random oscillations in the surrounding electromagnetic field. The best experimental determination of $$\mu_e$$ measures the small correction term $$\mu_e – \mu_B$$, enhancing the experiment’s sensitivity. The most recent results for the value of $$a$$ are:
$$ a \approx \frac{1}{860} $$
Precision and Accuracy
Since $$a$$ itself is a small correction term, the magnetic dipole moment of the electron is measured with an accuracy of about one part in $$10^{11}$$. The magnetic dipole moment of the electron is one of the most precisely determined quantities in physics, calculated correctly from quantum theory to within about one part in $$10^{10}$$.
The Interpretation of Quantum Mechanics
The Dual Nature of Electrons: Wave or Particle?
Quantum mechanics has been applied with great success in physics, but some of its ideas seem strange and counterintuitive. One such idea is the dual nature of the electron, which can exhibit both wave-like and particle-like properties.
Young’s Double-Slit Experiment
Young’s experiment, where a parallel beam of monochromatic light is passed through a pair of narrow parallel slits (Figure 5A), has an electron counterpart. In Young’s original experiment, the intensity of the light varies with direction after passing through the slits (Figure 5B). The intensity oscillates due to interference between the light waves emerging from the two slits, with the oscillation rate depending on the light’s wavelength and the slit separation. This creates a fringe pattern of alternating light and dark bands, modulated by the diffraction pattern from each slit. If one slit is covered, the interference fringes disappear, leaving only the diffraction pattern (shown as a broken line in Figure 5B).
Electron Double-Slit Experiment
Repeating Young’s experiment with electrons of the same momentum and using a closely spaced grid of electron detectors reveals the wave behavior of electrons. Scintillators are common electron detectors. When an electron passes through a scintillating material, like sodium iodide, the material produces a light flash, generating a voltage pulse that can be amplified and recorded. The pattern of electrons recorded matches that predicted for waves with wavelengths given by the de Broglie formula.
Single Electron Interference
If the experiment is conducted with a very weak electron source so that only one electron passes through the slits at a time, a single detector registers each electron’s arrival. This well-localized event is characteristic of a particle. Repeating the experiment many times and plotting a graph with detector position on one axis and the number of electrons on the other yields an oscillating interference pattern identical to Figure 5B. The intensity function in the figure is proportional to the probability of the electron’s direction after passing through the slits, matching $$ \Psi^2 $$, where $$ \Psi $$ is the solution of the time-independent Schrödinger equation for this experiment.
Which Slit?
If one slit is covered, the fringe pattern disappears and is replaced by a single-slit diffraction pattern. However, if the electron is a particle, it seems reasonable to assume it passes through only one slit. To determine which slit, the apparatus can be modified by placing a thin wire loop around each slit. When an electron passes through a loop, it generates a small electric signal, indicating the slit it passed through. However, this modification causes the interference fringe pattern to disappear, replaced by the single-slit diffraction pattern. Since both slits are needed for the interference pattern and it is impossible to know which slit the electron passed through without destroying the pattern, one must conclude that the electron goes through both slits simultaneously.
Wave-Particle Duality
The experiment demonstrates the electron’s wave and particle properties. The wave property predicts the probability of the electron’s travel direction before detection, while detection in a specific place shows particle properties. Therefore, the electron is neither purely a wave nor a particle. It exhibits wave or particle properties depending on the measurement type. Thus, one cannot speak of the intrinsic properties of an electron alone but must consider the properties of the electron and the measuring apparatus together.
Hidden Variables in Quantum Mechanics
Fundamental Randomness
A core concept in quantum mechanics is randomness or indeterminacy. Generally, the theory predicts only the probability of a specific result. Consider the example of radioactivity: imagine a box of atoms with identical nuclei capable of decaying by emitting an alpha particle. Over a given time interval, a certain fraction will decay. While quantum mechanics can predict this fraction precisely, it cannot determine which particular nuclei will decay. The theory asserts that at the beginning of the time interval, all nuclei are in an identical state, and the decay process is entirely random.
In contrast, many processes in classical physics appear random only because of our ignorance of initial conditions and the complexity of the necessary calculations. For instance, in a roulette wheel spin, knowing the exact position and speed of the wheel and other parameters could, in theory, allow us to predict the winning number. However, in quantum mechanics, randomness is asserted to be fundamentally intrinsic. Even if one nucleus decays while another does not, both were initially in the same state.
The Hidden Variable Hypothesis
Many eminent physicists, including Albert Einstein, rejected this indeterminacy, proposing instead that there must be some unknown property—a hidden variable—that differentiates the nuclei. If these hidden variables existed, it would restore determinacy to physics, allowing us to predict which nuclei would decay. Louis de Broglie, David Bohm, and others have attempted to construct theories based on hidden variables. However, these theories are often complicated and contrived. For example, in the double-slit experiment, such theories would require postulating a special force acting on the electron only when both slits are open, making them less attractive and garnering little support among physicists.
The Copenhagen Interpretation
The orthodox view of quantum mechanics, known as the Copenhagen interpretation, was primarily developed by Niels Bohr in Copenhagen. This interpretation emphasizes the importance of basing theory on what can be observed and measured experimentally, rejecting the idea of hidden variables as unmeasurable quantities. According to the Copenhagen interpretation, the observed indeterminacy in nature is fundamental and does not reflect an inadequacy in current scientific knowledge. We should accept this indeterminacy without trying to “explain” it and explore its consequences.
Indeterminacy and Free Will
Attempts have been made to link the indeterminacy of quantum mechanics with the existence of free will. However, this connection is tenuous. Free will implies rational thought and decision-making, while the indeterminacy in quantum mechanics stems from intrinsic randomness.
The Einstein-Podolsky-Rosen Paradox
The EPR Thought Experiment
In 1935, Einstein, Boris Podolsky, and Nathan Rosen proposed a thought experiment to measure position and momentum in a pair of interacting systems. Using conventional quantum mechanics, they derived startling results, concluding that the theory does not provide a complete description of physical reality. Their results, though paradoxical, were based on impeccable reasoning. However, their conclusion of the theory’s incompleteness does not necessarily follow. David Bohm later simplified their experiment while retaining its core reasoning.
Spin and Angular Momentum
The proton, like the electron, has spin $$ \frac{1}{2} $$. Thus, no matter what direction is chosen for measuring the component of its spin angular momentum, the values are always $$ +\frac{\hbar}{2} $$ or $$ -\frac{\hbar}{2} $$. Consider a system consisting of a pair of protons in close proximity with a total angular momentum of zero. If the angular momentum component of one proton is $$ +\frac{\hbar}{2} $$ along any direction, the other proton’s component in the same direction must be $$ -\frac{\hbar}{2} $$.
Suppose the protons move in opposite directions until they are far apart. The total angular momentum of the system remains zero. If the component of angular momentum along the same direction for each proton is measured, the results are equal and opposite. Thus, after measuring one proton, the other’s value can be predicted, making the second measurement unnecessary. Measuring a quantity changes the system’s state, so if measuring $$ S_x $$ (the x-component of angular momentum) for proton 1 produces $$ S_x = +\frac{\hbar}{2} $$, proton 1’s state corresponds to $$ S_x = +\frac{\hbar}{2} $$, and proton 2’s state corresponds to $$ S_x = -\frac{\hbar}{2} $$.
Quantum vs. Classical Descriptions
In classical physics, a system is assumed to possess the measured quantity beforehand. The measurement reveals the preexisting state without disturbing the system. In quantum mechanics, the system does not possess components of angular momentum before measurement, making the quantities hidden variables if they did exist.
Conclusion of the EPR Paradox
Einstein and his collaborators believed the conclusion that measuring one proton’s state determines the other, regardless of distance, was false. They concluded that quantum mechanics must be incomplete and that a correct theory would include hidden variables to restore classical determinism.
This paradox highlights the essential difference between quantum and classical outlooks, emphasizing the intrinsic indeterminacy and interconnectedness of quantum states over vast distances, a phenomenon later explored in the concept of quantum entanglement.
Testing Quantum Mechanics: Bell’s Inequality and Correlated Protons
Correlation Measurements and Bell’s Theorem
To determine if nature behaves as quantum mechanics predicts, one can measure the components of angular momenta for two protons along different directions with an angle $$ \theta $$ between them. A measurement on one proton yields either $$ +\frac{\hbar}{2} $$ or $$ -\frac{\hbar}{2} $$. The experiment involves measuring correlations between these plus and minus values for pairs of protons at a fixed $$ \theta $$ and repeating the measurements for various values of $$ \theta $$ (see Figure 6).
The interpretation of these results hinges on an important theorem by John Stewart Bell. Bell began by assuming the existence of some form of hidden variable that would determine whether the measured angular momentum gives a plus or minus result. He also assumed locality—that is, the idea that a measurement on one proton (i.e., the choice of measurement direction) cannot affect the result of the measurement on the other proton. Both assumptions align with classical, commonsense ideas.
Bell demonstrated that these assumptions lead to a specific relationship, known as Bell’s Inequality, for the correlation values. Experiments conducted in several laboratories using photons instead of protons (with similar analysis) have shown that Bell’s inequality is violated. The observed results agree with quantum mechanics and cannot be explained by a hidden variable theory based on locality. Thus, it must be concluded that the two protons are a correlated pair and a measurement of one affects the state of both, regardless of the distance separating them.
Instantaneous Effects and Relativity
It is noteworthy that the effect on the state of proton 2 following a measurement on proton 1 is believed to be instantaneous, occurring before a light signal initiated by the measurement at proton 1 can reach proton 2. This was demonstrated in 1982 by Alain Aspect and his coworkers in Paris, who designed an ingenious experiment. They measured the correlation between the two angular momenta within a very short time interval using a high-frequency switching device. The interval was shorter than the time required for a light signal to travel between the two measurement positions.
According to Einstein’s Special Theory of Relativity, no message can travel faster than the speed of light. Therefore, there is no way the information about the measurement direction on the first proton could reach the second proton before its measurement is made. This seemingly paradoxical result underscores the non-local nature of quantum mechanics.
These findings highlight the fundamental differences between classical and quantum physics. Quantum mechanics suggests that nature does not adhere to classical ideas of locality and determinism, as demonstrated by the violation of Bell’s Inequality. Instead, quantum entities exhibit a degree of interconnectedness that defies classical intuition, confirming the predictions of quantum theory.
Measurement in Quantum Mechanics
The Schrödinger Equation and Wave Function Collapse
Schrödinger’s time-dependent wave equation provides an exact method for determining how the wave function varies over time for a given physical system in a given environment. According to the Schrödinger equation, the wave function evolves in a strictly determinate manner. However, in the axiomatic approach to quantum mechanics, a measurement causes an abrupt and discontinuous change in the wave function.
Before measurement, the wave function $$\Psi$$ is a mixture of states $$\psi$$ as indicated in equation ($$\Psi = c_1 \psi_1 + c_2 \psi_2 + \cdots + c_N \psi_N$$). The measurement changes $$\Psi$$ from a mixture of $$\psi$$ states to a single $$\psi$$ state. This change, known as the collapse or reduction of the wave function, is unpredictable. Starting with the same $$\Psi$$, represented by the right-hand side of the equation ($$\Psi = c_1 \psi_1 + c_2 \psi_2 + \cdots + c_N \psi_N$$), the end result can be any one of the individual $$\psi$$ states.
The Schrödinger equation, which provides a smooth and predictable variation of $$\Psi$$, applies between measurements. However, the measurement process itself cannot be described by the Schrödinger equation; it is somehow separate. This appears unsatisfactory because a measurement is a physical process and ought to be described by the Schrödinger equation like any other physical process.
Quantum and Classical Worlds
The difficulty arises from the fact that quantum mechanics applies to microscopic systems containing one or a few electrons, protons, or photons. Measurements, however, are made with large-scale objects (e.g., detectors, amplifiers, and meters) in the macroscopic world, which obeys the laws of classical physics. Thus, another way of formulating the question of what happens during a measurement is to ask how the microscopic quantum world relates and interacts with the macroscopic classical world. More specifically, one can ask how and at what point in the measurement process the wave function collapses. So far, there are no satisfactory answers, although several schools of thought exist.
The Role of the Observer
One approach emphasizes the role of a conscious observer in the measurement process, suggesting that the wave function collapses when the observer reads the measuring instrument. However, bringing the conscious mind into the measurement problem raises more questions than it answers.
The Copenhagen Interpretation
The Copenhagen interpretation of the measurement process, essentially pragmatic, distinguishes between microscopic quantum systems and macroscopic measuring instruments. The initial object or event—e.g., the passage of an electron, photon, or atom—triggers the classical measuring device into giving a reading. Somewhere along this chain of events, the result of the measurement becomes fixed (i.e., the wave function collapses). This interpretation does not answer the fundamental question but suggests not to worry about it, a view probably held by most practicing physicists.
Irreversibility and Measurement
A third school of thought notes that an essential feature of the measurement process is irreversibility. This contrasts with the behavior of the wave function when it varies according to the Schrödinger equation, which in principle can be reversed by an appropriate experimental arrangement. However, once a classical measuring instrument has given a reading, the process is not reversible. It is possible that the key to understanding the measurement process lies here.
The Schrödinger equation applies only to relatively simple systems. Assuming the same equation applies to the large and complex system of a classical measuring device is an enormous extrapolation. It may be that the appropriate equation for such a system has features that produce irreversible effects (e.g., wave function collapse) different from those for a simple system.
The Many-Worlds Interpretation
The many-worlds interpretation, proposed by Hugh Everett III in 1957, suggests that when a measurement is made for a system where the wave function is a mixture of states, the universe branches into multiple non-interacting universes. Each possible outcome of the measurement occurs in a different universe. For example, if $$ S_x = \frac{1}{2} $$ is the result of a Stern-Gerlach measurement on a silver atom, there is another universe identical to ours except that the result of the measurement is $$ S_x = -\frac{1}{2} $$. Although this model solves some measurement problems, it has few adherents among physicists.
Various ways of understanding the measurement process lead to the same experimental consequences, making it difficult to distinguish between them on scientific grounds. One interpretation may be preferred for its plausibility, elegance, or economy of hypotheses, but these are matters of individual taste. Whether a satisfactory quantum theory of measurement will emerge, distinguished by its verifiable predictions, remains an open question.
