Theoretical and experimental justification for the Schrödinger equation

From Wikipedia, the free encyclopedia

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.

1 Classical electromagnetic waves
2 Quantum interpretation of classical electrodynamics
- 2.1 Energy, momentum, and angular momentum of photons
- 2.2 Probability for a single photon
  - 2.2.1 Probability amplitudes
3 de Broglie waves
- 3.1 de Broglie hypothesis
  - 3.1.1 Rest frame
  - 3.1.2 Frame of the observer
    - 3.1.2.1 Phase of the observer clock
    - 3.1.2.2 Phase of the observer wave
4 Bohr atom
5 Schrödinger equation
6 See also
7 References

[edit] Classical electromagnetic waves

Main article: Electromagnetic wave equation

[edit] Nature of the photon

Main article: 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 field vectors. 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, Maxwells 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 macroscopic objects, failed to describe very small objects such as electrons. Abductive reasoning was 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.

[edit] Plane sinusoidal waves

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

[edit] Electromagnetic wave equation

Main article: Electromagnetic wave equation

The electromagnetic 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 field E or the magnetic field B, takes the form:

$\nabla^2 \mathbf{E} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{E} \over \partial t^2} \ \ = \ \ 0$

$\nabla^2 \mathbf{B} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{B} \over \partial t^2} \ \ = \ \ 0$

where c is the speed of light in 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

$\nabla \times \mathbf{B} = {1 \over c} \frac{ \partial \mathbf{E}} {\partial t}$ .

[edit] Plane wave solution of the electromagnetic wave equation

Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarised wave propagating from left to right.

The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is (cgs units and SI units)

$\mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\zeta \rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \} \equiv \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\phi \rangle \right \}$

for the electric field and

$\mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t )$

for the magnetic field, where k is the wavenumber,

$\omega_{ }^{ } = c k$

is the angular frequency of the wave, and $c$ is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions. The quantity $\mid \mathbf{E} \mid$ is the amplitude of the wave.

Here

$|\zeta \rangle \equiv \begin{pmatrix} \zeta_x \\ \zeta_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix}$

is the Jones vector in the x-y plane. The notation for this vector is the bra-ket notation of 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 $\theta,\; \alpha_x,\; \mbox{and} \; \alpha_y$ are the angle the magnitude of the electric field makes with the x axis and the two initial phases of the wave, respectively.

The quantity

$|\phi \rangle = \exp \left [ i \left ( kz-\omega t \right ) \right ] |\zeta \rangle$

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.

[edit] Energy, momentum, and angular momentum of electromagnetic waves

[edit] Energy density of classical electromagnetic waves

[edit] Energy in a plane wave

Main article: Energy density

The energy per unit volume in classical electromagnetic fields is (cgs units)

$\mathcal{E}_c = \frac{1}{8\pi} \left [ \mathbf{E}^2( \mathbf{r} , t ) + \mathbf{B}^2( \mathbf{r} , t ) \right ]$ .

For a plane wave, this becomes

$\mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi}$

where the energy has been averaged over a wavelength of the wave.

[edit] Fraction of energy in each component

The fraction of energy in the x component of the plane wave is

$f_x = \frac{ \mid \mathbf{E} \mid^2 \cos^2\theta }{ \mid \mathbf{E} \mid^2 } = \phi_x^*\phi_x$

with a similar expression for the y component.

The fraction in both components is

$\phi_x^*\phi_x + \phi_y^*\phi_y = \langle \phi | \phi\rangle = 1$ .

[edit] Momentum density of classical electromagnetic waves

The momentum density is given by the Poynting vector

$\boldsymbol { \mathcal{P}} = {1 \over 4\pi c } \mathbf{E}( \mathbf{r}, t ) \times \mathbf{B}( \mathbf{r}, t )$ .

For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:

$\mathcal{P} c = \mathcal{E}_c$ .

The momentum density has been averaged over a wavelength.

[edit] Angular momentum density of classical electromagnetic waves

The angular momentum density is

$\boldsymbol { \mathcal{L} } = \mathbf{r} \times \boldsymbol { \mathcal{P} } = {1 \over 4\pi c } \mathbf{r} \times \left [ \mathbf{E}( \mathbf{r}, t ) \times \mathbf{B}( \mathbf{r}, t ) \right ]$ .

For a sinusoidal plane wave the angular momentum is in the z direction and is given by

$\mathcal{L} = { {\mid \mathbf{E} \mid^2} \over {8\pi\omega} } \left ( \mid \langle R | \psi\rangle \mid^2 - \mid \langle L | \phi\rangle \mid^2 \right ) = { 1 \over \omega } \mathcal{E}_c \left ( \mid \phi_R \mid^2 - \mid \phi_L \mid^2 \right )$

where again the density is averaged over a wavelength. Here right and left circularly polarized unit vectors are defined as

$|R\rangle \equiv {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}$

and

$|L\rangle \equiv {1 \over \sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}$ .

[edit] 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 $(t + τ)$ as

$| \phi (t + \tau) \rangle = \hat{U}(\tau) | \phi (t) \rangle$ .

To conserve energy in the wave we require

$\langle \phi (t + \tau)| \phi (t + \tau) \rangle = \langle \phi(t) | \hat{U}^{\dagger}(\tau) \hat{U}(\tau) | \phi(t) \rangle = \langle \phi(t) | \phi(t) \rangle = 1$

where $U^{\dagger}$ is the adjoint of U, the complex conjugate transpose of the matrix.

This implies that a transformation that conserves energy must obey

$\hat{U}^{\dagger} \hat{U} = I$

where I is the identity operator and U is called a unitary operator. The unitary property is necessary to ensure energy conservation in state transformations.

[edit] Hermitian operators and energy conservation

If $τ$ is an infinitesimal real quantity $d t$ , 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

$\hat{U} \approx I + i\hat{H}\tau$

and the adjoint by

$\hat{U}^{\dagger} \approx I - i\hat{H}^{\dagger}\tau$ .

Energy conservation then requires

$I = \hat{U}^{\dagger} \hat{U} \approx \left ( I - i\hat{H}^{\dagger}\tau \right ) \left ( I + i\hat{H}\tau \right ) \approx I - i\hat{H}^{\dagger}\tau + i\hat{H}\tau + \hat{H}^\dagger \hat{H} \tau^2$ .

Since $τ$ is infinitesimal, which means that $τ 2$ may be neglected with respect to $τ$ , the last term can be omitted. Further, if H is equal to its adjoint:

$\hat{H} = \hat{H}^{\dagger}$ ,

it follows that (for infinitesimal translations in time $\tau= dt\,$ )

$\hat{U}^{\dagger} \hat{U} = I$ ,

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

$|\phi (t+dt) \rangle - |\phi (t) \rangle = i\hat{H} |\phi (t)\rangle$ .

Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.

[edit] Quantum interpretation of classical electrodynamics

Main article: Photon

The treatment to this point has been classical. It is a testament, however, to the generality of Maxwell's equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinperpretation is that the state vectors

$\mid \phi \rangle$

in the classical description become quantum state vectors in the description of photons.

[edit] Energy, momentum, and angular momentum of photons

[edit] Energy

Max Planck presents Albert Einstein with the Max-Planck medal, Berlin June 28, 1929. Planck received a Nobel Prize in 1918 and Einstein received a Nobel Prize in 1921 for their identification of the photon.

The reinterpretation is based on the experiments of Max Planck and 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

$\epsilon = \hbar \omega$

where $\hbar$ is an experimentally determined quantity known as Planck's constant. If there are $N$ photons in a box of volume $V$ , the energy in the electromagnetic field is

$N \hbar \omega$

and the energy density is

${N \hbar \omega \over V}$

The energy of a photon can be related to classical fields through the correspondence principle which states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large $N$ , the quantum energy density must be the same as the classical energy density

${N \hbar \omega \over V} = \mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi}$ .

The number of photons in the box is then

$N = \frac{V }{8\pi \hbar \omega}\mid \mathbf{E} \mid^2$ .

[edit] Momentum

The correspondence principle also determines the momentum and angular momentum of the photon. For momentum

$\mathcal{P}_c = {N \hbar \omega \over cV} = {N \hbar k \over V}$

which implies that the momentum of a photon is

$\hbar k$ .

[edit] Angular momentum and spin

Similarly for the angular momentum

$\mathcal{L} = { 1 \over \omega } \mathcal{E}_c \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right ) = { \hbar \over V } \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right )$

which implies that the angular momentum of the photon is

$l_z = \hbar \left ( \mid \psi_R \mid^2 - \mid \psi_L \mid^2 \right )$ .

the quantum interpretation of this expression is that the photon has a probability of $\mid \psi_R \mid^2$ of having an angular momentum of $\hbar$ and a probability of $\mid \psi_L \mid^2$ of having an angular momentum of $-\hbar$ . 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 $\pm \hbar$ .

[edit] Spin operator

The spin of the photon is defined as the coefficient of $\hbar$ in the angular momentum calculation. A photon has spin 1 if it is in the $| R \rangle$ state and -1 if it is in the $| L \rangle$ state. The spin operator is defined as the outer product

$\hat{S} \equiv |R\rangle \langle R | - |L\rangle \langle L | = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ .

The eigenvectors of the spin operator are $|R\rangle$ and $|L\rangle$ with eigenvalues 1 and -1, respectively.

The expected value of a spin measurement on a photon is then

$\langle \psi |\hat{S} |\psi\rangle = \mid \psi_R \mid^2 - \mid \psi_L \mid^2$ .

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.

[edit] 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 double-slit experiment:

“	Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.	”
—Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1

[edit] Probability amplitudes

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 amplitude contains 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.

[edit] de Broglie waves

Louis de Broglie. De Broglie received the Nobel Prize in Physics in 1929 for his identification of waves with particles.

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 consistent with 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.

[edit] de Broglie hypothesis

Main article: De Broglie hypothesis

De Broglie supposed that every particle was associated with both a particle and a wave. The angular frequency $ω$ and wavenumber $k$ of the wave was related to the energy E and momentum p of the particle by

$E=\hbar \omega$

and

$p=\hbar k$ .

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.

[edit] Rest frame

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

$E_0 = mc^2 = \hbar \omega_0$

and

$p_0 = 0 =\hbar k_0$

where m is the rest mass of the particle.

This describes a wave of infinite wavelength and infinite phase velocity

$v_{\phi} = {\omega_0 \over k_0 }$ .

The wave may be written as proportional to

$cos(\omega_0^{ } t )$ .

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.

[edit] 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.

[edit] 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

$\omega_c = {\omega_0 \over \gamma}$

where

$\gamma = {1 \over \sqrt{1-{v^2 \over c^2} } }$

is a factor that describes time dilation of the particle clock as observed by the observer.

The phase of the observer clock is

$\omega_c t = {\omega_0 \over \gamma} ( \gamma t_0 ) = \omega_0 t_0$

where $t 0$ is time measured in the particle frame. Both the observer clock and the particle clock agree on the phase.

[edit] Phase of the observer wave

The frequency and wavenumber of the wave in the observer frame is given by

$E = \gamma mc^2 = \hbar \omega = \gamma \hbar \omega_0$

and

$p = \gamma m v = \hbar k$

with a phase velocity

$v_{\phi} = {\omega \over k} = {E \over p} = {c^2 \over v}$ .

The phase of the wave in the observer frame is

$\omega t - k z = \omega t - {\omega \over v_{\phi}} vt = \omega t \left ( 1- {v^2 \over c^2} \right ) = {\omega t \over \gamma^2} = {1 \over \gamma^2}{\gamma mc^2 \over \hbar} (\gamma t_0 ) = \omega_0 t_0 = \omega_c t$ .

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.

[edit] Bohr atom

Main article: Bohr atom

Niels Bohr. In 1922 the Nobel Prize in Physics was awarded to Niels Bohr for his contributions to the understanding of quantum mechanics.

[edit] Inconsistency of observation with classical physics

The de Broglie hypothesis helped resolve outstanding issues in atomic physics. Classical physics was 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 mechanics that 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.

[edit] Balmer series

Main article: Balmer series

The Balmer series identifies those frequencies of light that can be emitted from an excited hydrogen atom:

$\hbar \omega_n = R \left(\frac{1}{2^2} - \frac{1}{n^2}\right) \quad n=3,4,5,...$

where R is known at the Rydberg constant and is equal to 13.6 electron volts.

[edit] 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 mechanics do 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
${L} = n \hbar$
Where n = 1,2,3,… and is called the principal quantum number.

[edit] Implications of the Bohr model

In a circular orbit the attractive force balances the centrifugal force of the electron

${mv^2 \over r } = {e^2 \over r^2}$

where e is the charge on the electron or proton, m is the mass of the electron, v is the speed of the electron, and r is the radius of the orbit.

The energy of the orbiting electron is

$E = {1 \over 2} m v^2 - {e^2 \over r} = -{1\over 2} m v^2 = -{1\over 2} {e^2\over r}$

where the last two relations follow from the centrifugal force expression.

The angular momentum assumption of the Bohr model implies

$L = mvr = n \hbar$

which implies that, when combined with the centrifugal force equation, the radius of the orbit is given by

$r= {n^2 \hbar^2 \over m e^2 }$ .

This implies, from the energy equation,

$E_n = -{1\over 2} {e^2\over r} = -{1\over 2} \left ( {me^4\over \hbar^2} \right ) {1\over n^2}$ .

The difference between energy levels recovers the Balmer series.

[edit] 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

$p = mv = \hbar k$

as postulated by the de Broglie hypothesis, then the angular momentum is given by

$L = mvr = \hbar k r = \hbar \left ( {2\pi \over \lambda} \right ) r$

where $λ$ is the wavelength of the electron wave.

If only standing electron waves are permitted in the atom then only orbits with radii equal to integral numbers of wavelengths are allowed:

$\lambda = {2 \pi r \over n}$ .

This implies that allowed orbits have angular momentum

$L=n\hbar$

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.

[edit] 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.

[edit] Schrödinger equation

Main article: Schrödinger equation

Erwin Schrödinger. In 1933 Erwin Schrödinger received the Nobel Prize in Physics for his development of the equation that bears his name.

[edit] Analogy with photon dynamics

The dynamics of a photon is given by

$|\phi (t+dt) \rangle - |\phi (t) \rangle = - i\hat{H} dt |\phi (t)\rangle$

where H is a Hermitian operator determined by Maxwell's equations.

Erwin Schrödinger assumed that the dynamics for massive particles were of the same form as photon dynamics.

$|\psi (t+dt) \rangle - |\psi (t) \rangle = - i\hat{H} dt |\psi (t)\rangle$

where $|\psi (t) \rangle$ is the state vector for the particle and H is now an unknown Hermitian operator to be determined.

[edit] 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 $d x$ and assigned a component of the state vector to each bin

$|\psi\rangle \equiv \begin{pmatrix} \vdots \\ \psi_{i-1} \\ \psi_i \\ \psi_{i+1} \\ \vdots \end{pmatrix}$ .

[edit] Transition amplitudes

Schrödinger assumed that H was composed of probability amplitudes for transitioning from one bin $i$ to a neighboring bin $i \pm 1$ of the form

$h_{i \pm 1, i}^{ } \; \psi_i(t)$ .

[edit] Schrödinger equation in one dimension

Probability densities for the electron at different quantum numbers in the hydrogen atom.

Assigning a transition probability for every way in which a particle can enter or leave a bin, taking the limits of infinitesimal dt and dx, and equating the plane wave solution with the de Broglie solution leads to the Schrödinger equation in one dimension:

$-\frac{\hbar^2}{2 m} \frac{d^2 \psi (x, t)}{dx^2} + U(x) \psi (x, t) = i\hbar {\partial \over \partial t} \psi (x, t)$

where U is the classical potential energy and

$\psi (x, t) \equiv {1\over \sqrt{dx} } \psi_i (t)$

and

$1 = \int_{-\infty}^{\infty} \psi^* (x, t) \psi (x, t) dx$ .

[edit] Schrödinger equation in three dimensions

In three dimensions the Schrödinger equation becomes

$-\frac{\hbar^2}{2 m} { \nabla^2 \psi} + U \psi = i\hbar {\partial \over \partial t} \psi$

[edit] Generalised Schrödinger equation

In the state vector formulation of quantum mechanics, Schrödinger's equation is generalised to

$i \hbar \frac{d}{dt} | \psi \rangle = \hat{H} | \psi \rangle$

where $| \psi \rangle$ is the state vector and Ĥ is the Hamiltonian operator.

[edit] Hydrogen atom

The solution for the hydrogen atom describes standing waves of energy exactly given by the Balmer series. This was a spectacular verification of the wave-like nature of matter.

[edit] See also

[edit] References

Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X.
Baym, Gordon (1969). Lectures on Quantum Mechanics. W. A. Benjamin. ISBN 68-56111.
Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, Fourth Edition. Oxford. ISBN 0-19-851208-2.

v • d • e

General subfields within physics

Classical mechanics · Electromagnetism · Thermodynamics · General relativity · Quantum mechanics

Particle physics · Thermal physics · Condensed matter physics · Atomic, molecular, and optical physics

Retrieved from "http://en.wikipedia.org../../../t/h/e/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation_a90b.html"

Categories: Partial differential equations | Quantum mechanics | Equations | Fundamental physics concepts