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### 6.1.1.1 Schrödinger Equation

The time-dependent single-particle Schrödinger equation is

 (6.1)

Here is the complex wave function. The Hamiltonian operator 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 the Hamiltonian is

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

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

 (6.4)

In one-dimensional position space the position operator is given by (i.e., multiplication of by ). The momentum operator is then given by . With this the transient Schrödinger equation for the electron wave function on the real line reads (with constant mass)

 (6.5)

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

 (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 ,

 (6.7)

or with 6.2

 (6.8)

The inner product in the Hilbert space determines the probabilistic structure. Observables are represented by self-adjoint operators. The inner product of two vectors is given by

 (6.9)

where we used Dirac's bra-ket'' notation. The eigenvectors of the position operator are denoted by , we write for the eigenvectors of the momentum operator. The wave functions and are then recovered as

 (6.10)

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

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

 (6.11)

Note that for non self-adjoint operators the notation is ambiguous.

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