- Theoretical and experimental justification for the Schrödinger equation
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the
Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics determined by Maxwell's equations, as an analogue for all types of particles.
Classical electromagnetic waves
Nature of the photon
The quantum particle of light is called a
photon. Photons have both a wave-like and a particle-like nature. In other words, photons can appear to be particles in some experiments and waves in other experiments. The dynamics of photons are completely determined by Maxwell's equations, the classical description of electrodynamics. In the absence of sources, Maxwell's equations can be written as wave equations in the electric and magnetic fieldvectors. Maxwell's equations thus describe the wave-like properties of the photon. When the fields are interpreted as being proportional to probability amplitudes for finding a photon particle in a particular state, Maxwell's equations also describe the particle-like properties of the photon.
Maxwell's equations were completely known by the latter part of the nineteenth century. The dynamical equations for the quantum photon were, therefore, well-known long before the discovery of the photon. This is not true for other particles such as the
electron. It was surmised from the interaction of light with atoms that electrons also had both a particle-like and a wave-like nature. Newtonian mechanics, a description of the particle-like behavior of macroscopicobjects, failed to describe very small objects such as electrons. Abductive reasoningwas performed to obtain the dynamics of massive (particles with mass) objects such as electrons. The electromagnetic wave equation, the equation that described the dynamics of the photon, were used as a prototype for discovering the Schrödinger equation, the equation that describes the wave-like and particle-like dynamics of nonrelativistic massive particles.
Plane sinusoidal waves
Electromagnetic wave equation
The electromagnetic wave equation describes the propagation of electromagnetic waves through a medium or in a
vacuum. The homogeneous form of the equation, written in terms of either the electric fieldE or the magnetic fieldB, takes the form:
where "c" is the
speed of lightin the medium. In a vacuum, c = 2.998 x 108 meters per second, which is the speed of light in free space.
The magnetic field is related to the electric field through Faraday's law (
Plane wave solution of the electromagnetic wave equation
for the electric field and
for the magnetic field, where k is the
angular frequencyof the wave, and is the speed of light. The hats on the vectors indicate unit vectorsin the x, y, and z directions. The quantity is the amplitudeof the wave.
Jones vectorin the x-y plane. The notation for this vector is the bra-ket notationof Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector. The angles are the angle the electric field makes with the x axis and the two initial phases of the wave, respectively.
is the state vector of the wave. It describes the polarization of the wave and the spatial and temporal functionality of the wave. This quantity is also the quantum state of the photon.
Energy, momentum, and angular momentum of electromagnetic waves
Energy density of classical electromagnetic waves
Energy in a plane wave
The energy per unit volume in classical electromagnetic fields is (cgs units)
For a plane wave, this becomes
where the energy has been averaged over a wavelength of the wave.
Fraction of energy in each component
The fraction of energy in the x component of the plane wave is
with a similar expression for the y component.
The fraction in both components is
Momentum density of classical electromagnetic waves
The momentum density is given by the
For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:
The momentum density has been averaged over a wavelength.
Angular momentum density of classical electromagnetic waves
The angular momentum density is
For a sinusoidal plane wave the angular momentum is in the z direction and is given by
where again the density is averaged over a wavelength. Here right and left circularly polarized unit vectors are defined as
Unitary operators and energy conservation
A wave can be transformed by, for example, passing through a birefringent crystal or through slits in a
diffraction grating. We can define the transformation of the state from the state at time t to the state at time as
To conserve energy in the wave we require
where is the adjoint of U, the complex conjugate transpose of the matrix.
This implies that a transformation that conserves energy must obey:
where I is the identity operator and U is called a
unitary operator. The unitary property is necessary to ensure energy conservationin state transformations.
Hermitian operators and energy conservation
If is an infinitesimal real quantity , then the unitary transformation is very close to the identity matrix (the final state is very close to the initial state) and can be written :
and the adjoint by
The factor of i is introduced for convenience. With this convention, it will be shown that energy conservation requires H to be a Hermitian operator and that H is related to the energy of a particle.
Energy conservation requires
:.Since is infinitesimal, which means that may be neglectedwith respect to , the last term can be omitted. Further, if "H" is equal to itsadjoint::,it follows that (for infinitesimal translations in time ):,so that, indeed, energy is conserved.
Operators that are equal to their adjoints are called Hermitian or self-adjoint.
The infinitesimal translation of the polarization state is
Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator. While this derivation is classical, the concept of a Hermitian operator generating energy-conserving infinitesimal transformations forms an important basis for quantum mechanics. The derivation of the Schrödinger equation follows directly from this concept.
Quantum interpretation of classical electrodynamics
The treatment to this point has been classical. It is a testament, however, to the generality of
Maxwell's equationsfor electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinterpretation is that the state vectors
in the classical description become quantum state vectors in the description of photons.
Energy, momentum, and angular momentum of photons
The reinterpretation is based on the experiments of
Max Planckand the interpretation of those experiments by Albert Einstein. The important conclusion from these early experiments is that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relation
where is an experimentally determined quantity known as
Planck's constant. If there are photons in a box of volume , the energy in the electromagnetic field is
and the energy density is
The energy of a photon can be related to classical fields through the
correspondence principlewhich states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large , the quantum energy density must be the same as the classical energy density
The number of photons in the box is then
The correspondence principle also determines the momentum and angular momentum of the photon. For momentum
which implies that the momentum of a photon is
Angular momentum and spin
Similarly for the angular momentum
which implies that the angular momentum of the photon is
the quantum interpretation of this expression is that the photon has a probability of of having an angular momentum of and a probability of of having an angular momentum of . We can therefore think of the angular momentum of the photon being quantized as well as the energy. This has indeed been experimentally verified. Photons have only been observed to have angular momenta of .
The spin of the photon is defined as the coefficient of in the angular momentum calculation. A photon has spin 1 if it is in the state and -1 if it is in the state. The spin operator is defined as the
eigenvectors of the spin operator are and with eigenvalues 1 and -1, respectively.
The expected value of a spin measurement on a photon is then
An operator S has been associated with an observable quantity, the angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for angular momentum, but it is in general true for any observable quantity.
Probability for a single photon
There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation violates energy conservation. The latter interpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the
The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The state of the photon is proportional to the field. The probability itself is quadratic in the fields and consequently is also quadratic in the quantum state of the photon. In quantum mechanics, therefore, the state or
probability amplitudecontains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: [The following quote is from Baym, Chapter 1]
# The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. ...
# The amplitude for a process that can take place in place in one of several indistinguishable ways is the sum of amplitudes for each of the individual ways. ...
# The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.
de Broglie waves
In 1923 Louis de Broglie addressed the question of whether all particles can have both a wave and a particle nature similar to the photon. Photons differ from many other particles in that they are massless and travel at the speed of light. Specifically de Broglie asked the question of whether a particle that has both a wave and a particle associated with it is
consistentwith Einstein's two great 1905 contributions, the special theory of relativity and the quantization of energy and momentum. The answer turned out to be positive. The wave and particle nature of electrons was experimentally observed in 1927, two years after the discovery of the Schrödinger equation.
de Broglie hypothesis
De Broglie supposed that every particle was associated with both a particle and a wave. The angular frequency and wavenumber of the wave was related to the energy E and momentum p of the particle by
The question reduces to whether every observer in every inertial reference frame can agree on the phase of the wave. If so, then a wave-like description of particles may be consistent with special relativity.
First consider the rest frame of the particle. In that case the frequency and wavenumber of the wave are related to the energy and momentum of the particles properties by
where m is the rest mass of the particle.
This describes a wave of infinite wavelength and infinite
The wave may be written as proportional to
This, however, is also the solution for a
simple harmonic oscillator, which can be thought of as a clock in the rest frame of the particle. We can imagine a clock ticking at the same frequency as the wave is oscillating. The phases of the wave and the clock can be synchronized.
Frame of the observer
It is shown that the phase of the wave in an observer frame is the same as the phase of the wave in a particle frame, and also the same as clocks in the two frames. There is, therefore, consistency of both a wave-like and a particle-like picture in special relativity.
Phase of the observer clock
In the frame of an observer moving at relative speed v with respect to the particle, the particle clock is observed to tick at a frequency
is a factor that describes
time dilationof the particle clock as observed by the observer.
The phase of the observer clock is
where is time measured in the particle frame. Both the observer clock and the particle clock agree on the phase.
Phase of the observer wave
The frequency and wavenumber of the wave in the observer frame is given by
with a phase velocity
The phase of the wave in the observer frame is
The phase of the wave in the observer frame is the same as the phase in the particle frame, as the clock in the particle frame, and the clock in the observer frame. A wave-like picture of particles is consistent with special relativity.
Inconsistency of observation with classical physics
The de Broglie hypothesis helped resolve outstanding issues in atomic physics.
Classical physicswas unable to explain the observed behaviour of electrons in atoms. Specifically, accelerating electrons emit electromagnetic radiation according to the Larmor formula. Electrons orbiting a nucleus should lose energy to radiation and eventually spiral into the nucleus. This is not observed. Atoms are stable on timescales much longer than predicted by the classical Larmor formula.
Also, it was noted that excited atoms emit radiation with discrete frequencies. Einstein used this fact to interpret discrete energy packets of light as, in fact, real particles. If these real particles are emitted from atoms in discrete energy packets, however, must the emitters, the electrons, also change energy in discrete energy packets? There is nothing in
Newtonian mechanicsthat explains this.
The de Broglie hypothesis helped explain these phenomena by noting that the only allowed states for an electron orbiting an atom are those that allow for standing waves associated with each electron.
The Balmer series identifies those frequencies of light that can be emitted from an excited hydrogen atom:
where R is known at the
Rydberg constantand is equal to 13.6 electron volts.
Assumptions of the Bohr model
The Bohr model, introduced in 1913, was an attempt to provide a theoretical basis for the Balmer series. The assumptions of the model are:
# The orbiting electrons existed in circular orbits that had discrete quantized energies. That is, not every orbit is possible but only certain specific ones.
# The laws of
classical mechanicsdo not apply when electrons make the jump from one allowed orbit to another.
# When an electron makes a jump from one orbit to another the energy difference is carried off (or supplied) by a single quantum of light (called a
photon) which has an energy equal to the energy difference between the two orbitals.
# The allowed orbits depend on quantized (discrete) values of orbital
angular momentum, "L" according to the equation
Where "n" = 1,2,3,… and is called the
principal quantum number.
Implications of the Bohr model
In a circular orbit the
centrifugal forcebalances the attractive force of the electron
where m is the mass of the electron, v is the speed of the electron, r is the radius of the orbit and
where e is the charge on the electron or proton.
The energy of the orbiting electron is
which follows from the centrifugal force expression.
The angular momentum assumption of the Bohr model implies
which implies that, when combined with the centrifugal force equation, the radius of the orbit is given by
This implies, from the energy equation,
The difference between energy levels recovers the Balmer series.
De Broglie's contribution to the Bohr model
The Bohr assumptions recover the observed Balmer series. The Bohr assumptions themselves, however, are not based on any more general theory. Why, for instance, should the allowed orbits depend on the angular momentum? The de Broglie hypothesis provides some insight.
If we assume that the electron has a momentum given by
as postulated by the de Broglie hypothesis, then the angular momentum is given by
where is the wavelength of the electron wave.
If only standing electron waves are permitted in the atom then only orbits with perimeters equal to integral numbers of wavelengths are allowed:
This implies that allowed orbits have angular momentum
which is Bohr's fourth assumption.
Assumptions one and two immediately follow. Assumption three follows from energy conservation, which de Broglie showed was consistent the wave interpretation of particles.
Need for dynamical equations
The problem with the de Broglie hypothesis as applied to the Bohr atom is that we have forced a plane wave solution valid in empty space to a situation in which there is a strong attractive potential. We have not yet discovered the general dynamic equation for the evolution of electron waves. The Schrödinger equation is the immediate generalization of the de Broglie hypothesis and the dynamics of the photon.
Analogy with photon dynamics
The dynamics of a photon is given by
where H is a Hermitian operator determined by Maxwell's equations. The Hermiticity of the operator ensures that energy is conserved.
Erwin Schrödingerassumed that the dynamics for massive particles were of the same form as the energy-conserving photon dynamics.
where is the state vector for the particle and H is now an unknown Hermitian operator to be determined.
Particle state vector
Rather than polarization states as in the photon case, Schrödinger assumed the state of the vector depended on the position of the particle. If a particle lives in one spatial dimension, then he divided the line up into an infinite number of small bins of length and assigned a component of the state vector to each bin
The subscript j identifies the bin.
Matrix form and transition amplitudes
The transition equation can be written in matrix form as
The Hermition condition requires
Schrödinger assumed that probability could only leak into adjacent bins during the small time step dt. In other words, all components of H are zero except for transitions between neighboring bins
Moreover, it is assumed that space is uniform in that all transitions to the right are equal
The same is true for transitions to the left
The transition equation becomes
The first term on the right represents the movement of probability amplitude into bin j from the right. The second term represents leakage of probability from bin j to the right. The third term represents leakage of probability into bin j from the left. The fourth term represents leakage from bin j to the left. The final term represents any change of phase in the probability amplitude in bin j.
If we expand the probability amplitude to second order in the bin size and assume space is isotropic, the transition equation reduces to
chrödinger equation in one dimension
The transition equation must be consistent with the de Broglie hypothesis. In free space the probability amplitude for the de Broglie wave is proportional to
in the non-relativistic limit.
The de Broglie solution for free space is a solution of the transition equation if we require
The time derivative term in the transition equation can be identified with the energy of the de Broglie wave. The spatial derivative term can be identified with the kinetic energy. This suggests that the term containing is proportional to the potential energy. This yields the Schrödinger equation
where U is the classical potential energy and
chrödinger equation in three dimensions
In three dimensions the Schrödinger equation becomes
The solution for the hydrogen atom describes standing waves of energy exactly given by the Balmer series. This was a spectacular validation of the Schrödinger equation and of the wave-like nature of matter.
Path integral formulation
Photon dynamics in the double-slit experiment
*cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 047130932X
*cite book |author=Baym, Gordon |title=Lectures on Quantum Mechanics|publisher=W. A. Benjamin|year=1969|id=ISBN 68-56111
*cite book |author=Dirac, P. A. M. |title=The Principles of Quantum Mechanics, Fourth Edition|publisher=Oxford|year=1958|id=ISBN 0-19-851208-2
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