6.1.1.1 Schrödinger Equation

The time-dependent single-particle Schrödinger equation is

$$\imath \hbar \frac{\partial \psi}{\partial t} = H \psi .$$ (6.1)

Here $\psi(x,t)$ is the complex wave function. The Hamiltonian operator $H$ is a Hermitian (self-adjoint) linear operator acting on the state space. The Hamiltonian describes the total energy of the system.

For a particle with potential energy $V(x)$ the Hamiltonian is

$$H = \frac{P^2}{2m} + V .$$ (6.2)

If the effective mass $m$ is space dependent we choose the operator ordering

$$H = P\frac{1}{2m}P + V .$$ (6.3)

to get a self-adjoint Hamiltonian. In all our applications the effective mass is piecewise constant.

Position and momentum operators obey the canonical commutation relations

$$[ X^{i}, P^{j} ] = \imath \hbar \delta^{ij} .$$ (6.4)

In one-dimensional position space $L^2(\mathbb{R})$ the position operator $X$ is given by $x$ (i.e., multiplication of $\psi(x)$ by $x$). The momentum operator $P$ is then given by $-\imath \hbar \frac{\partial}{\partial x}$. With this the transient Schrödinger equation for the electron wave function $\psi(x,t)$ on the real line $\mathbb{R}$ reads (with constant mass)

$$\imath \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m^{*}} \psi'' + V(x,t) \psi .$$ (6.5)

In the so called time-independent case the wave function is of the form

$$\psi(x,t) = e^{-\imath E t/\hbar} \psi(x) .$$ (6.6)

As time progresses, the state vectors change only by a complex phase and Schrödinger's equation becomes an eigenvalue equation for the Hamiltonian $H$,

$$H \psi = E \psi$$ (6.7)

or with 6.2

$$- \frac{\hbar^2}{2 m^{*}} \psi'' + V(x) \psi = E \psi .$$ (6.8)

The inner product in the Hilbert space determines the probabilistic structure. Observables are represented by self-adjoint operators. The inner product $\langle \psi \vert \phi \rangle$ of two vectors is given by

$$\langle \psi \vert \phi \rangle = \int \psi^{\star}(x) \phi(x) dx$$ (6.9)

where we used Dirac's ``bra-ket'' notation. The eigenvectors of the position operator are denoted by $\vert x\rangle$, we write $\vert p\rangle$ for the eigenvectors of the momentum operator. The wave functions $\psi(x)$ and $\phi(p)$ are then recovered as

$$\psi(x) = \langle x \vert \psi \rangle \quad \phi(p) = \langle p \vert \phi \rangle$$ (6.10)

for abstract wave vectors $\psi$, $\phi$. The relation beween space and momentum representation is a Fourier transformation

$$\phi(p) = \frac{1}{\sqrt{2 \pi \hbar}} \int dx \psi(x) e^{-\imath... ...ad \psi(x) = \frac{1}{\sqrt{2 \pi \hbar}} \int dp \phi(p) e^{\imath xp /\hbar}$$

Observables are repesented by self-adjoint operators. In the Dirac notation the expectation value of an observable $A$ given in state $\psi$ is denoted by

$$\bar A = \langle \psi \vert A \vert \psi \rangle = \langle \psi \vert A \psi \rangle = \langle A \psi \vert \psi \rangle .$$ (6.11)

Note that for non self-adjoint operators $O$ the notation $\langle \psi \vert O \vert \psi \rangle$ is ambiguous.