Materials Science and Engineering/Derivations/Quantum Mechanics

Schrodinger Equation

Schrödinger's equation follows very naturally from earlier developments:

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

$E=hf$

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 $\mathrm{\Psi }$ with properties:

$p=h/\lambda$, where $\lambda$ 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

$E=hf$

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

Expressed in terms of angular frequency $\omega =2\pi f$ and wavenumber $k=2\pi /\lambda$, with $\hslash =h/2\pi$ we get:

$E=\hslash \omega$

and

$𝐩=\hslash 𝐤$

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:

$\psi \approx e^{i(𝐤\cdot 𝐱-\omega t)}$

and to realize that since

$\frac{\partial }{\partial t}\psi =-i\omega \psi$

then

$E\psi =\hslash \omega \psi =i\hslash \frac{\partial }{\partial t}\psi$

and similarly since:

$\frac{\partial }{\partial x}\psi =i{k}_x\psi$

then

$p_x\psi =\hslash k_x\psi =-i\hslash \frac{\partial }{\partial x}\psi$

and hence:

$p_x^2\psi =-{\hslash }^2\frac{{\partial }^2}{\partial x^2}\psi$

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

$p^2\psi =(p_x^2+p_y^2+p_z^2)\psi =-{\hslash }^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2})\psi =-{\hslash }^2{\mathrm{\nabla }}^2\psi$

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

$E=\frac{p^2}{2m}+V$ (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:

$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }+V\mathrm{\Psi }$

Solution of the Time-Dependent Schrodinger Equation

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

$\mathrm{i}\hslash \frac{\partial }{\partial t}|{\psi }_n\left(t\right)⟩=E_n|{\psi }_n\left(t\right)⟩.$

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:

$|\psi \left(t\right)⟩={\mathrm{e}}^{-\mathrm{i}Et/\hslash }|\psi \left(0\right)⟩.$

It immediately follows that the probability amplitude,

$\psi (t{)}^{*}\psi (t)={\mathrm{e}}^{\mathrm{i}Et/\hslash }{\mathrm{e}}^{-\mathrm{i}Et/\hslash }\psi (0{)}^{*}\psi (0)=|\psi (0){|}^2,$

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 $\psi (t)$ are time-independent.

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

$|\psi \left(t\right)⟩=\sum _n{c}_n(t)|n⟩,H|n⟩=E_n|n⟩,\sum _n{\left|c_n\left(t\right)\right|}^2=1.$

(The last equation enforces the requirement that $|\psi \left(t\right)⟩$, 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

$\mathrm{i}\hslash \frac{\partial c_n}{\partial t}=E_n{c}_n\left(t\right).$

Therefore, if we know the decomposition of $|\psi \left(t\right)⟩$ into the energy basis at time $t=0$, its value at any subsequent time is given simply by

$|\psi \left(t\right)⟩=\sum _n{\mathrm{e}}^{-\mathrm{i}{E}_{n}t/\hslash }{c}_n\left(0\right)|n⟩.$

Note that when some values $c_n(0)$ are not equal to zero for differing energy values $E_n$, the left-hand side is not an eigenvector of the energy operator $H$. The left-hand is an eigenvector when the only $c_n(0)$-values not equal to zero belong the same energy, so that ${\mathrm{e}}^{-\mathrm{i}{E}_{n}t/\hslash }$ can be factored out. In many real-world application this is the case and the state vector $\psi (t)$ (containing time only in its phase factor) is then a solution of the time-independent Schrödinger equation.