(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

(6.11) |

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

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