Materials Science and Engineering/Derivations/Quantum Mechanics

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Schrodinger Equation[edit | edit source]

Schrödinger's equation follows very naturally from earlier developments:

In 1905, by considering the photoelectric effect, Albert Einstein had published his

formula for the relation between the energy E and frequency f of the quanta of radiation (photons), where h is Planck's constant.

In 1924 Louis de Broglie presented his de Broglie hypothesis which states that all particles (not just photons) have an associated wavefunction with properties:

, where is the wavelength of the wave and p the momentum of the particle.

De Broglie showed that this was consistent with Einstein's formula and special relativity so that

still holds, but now this is hypothesized to hold for all particles, not just photons anymore.

Expressed in terms of angular frequency and wavenumber , with we get:


where we have expressed p and k as vectors.

Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:

and to realize that since


and similarly since:


and hence:

so that, again for a plane wave, he got:

And by inserting these expressions into the Newtonian formula for a particle with total energy E, mass m, moving in a potential V:

(simply the sum of the kinetic energy and potential energy; the plane wave model assumed V = 0)

he got his famed equation for a single particle in the 3-dimensional case in the presence of a potential:

Solution of the Time-Dependent Schrodinger Equation[edit | edit source]

On inserting a solution of the time-independent Schrödinger equation into the full Schrödinger equation, we get

It is relatively easy to solve this equation. One finds that the energy eigenstates (i.e., solutions of the time-independent Schrödinger equation) change as a function of time only trivially, namely, only by a complex phase (waves)|phase:

It immediately follows that the probability amplitude,

is time-independent. Because of a similar cancellation of phase factors in bra and ket, all average (expectation) values of time-independent observables (physical quantities) computed from are time-independent.

Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors form a basis (linear algebra)|basis for the state space. We introduced here the short-hand notation . Then any state vector that is a solution of the time-dependent Schrödinger equation (with a time-independent ) can be written as a linear superposition of energy eigenstates:

(The last equation enforces the requirement that , like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the time-dependent Schrödinger equation in the left-hand side and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain

Therefore, if we know the decomposition of into the energy basis at time , its value at any subsequent time is given simply by

Note that when some values are not equal to zero for differing energy values , the left-hand side is not an eigenvector of the energy operator . The left-hand is an eigenvector when the only -values not equal to zero belong the same energy, so that can be factored out. In many real-world application this is the case and the state vector (containing time only in its phase factor) is then a solution of the time-independent Schrödinger equation.