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The time-dependent single-particle Schrödinger equation is
![$\displaystyle \imath \hbar \frac{\partial \psi}{\partial t} = H \psi .$](img308.png) |
(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
![$\displaystyle [ X^{i}, P^{j} ] = \imath \hbar \delta^{ij} .$](img315.png) |
(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)
![$\displaystyle \imath \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m^{*}} \psi'' + V(x,t) \psi .$](img322.png) |
(6.5) |
In the so called time-independent case the wave
function is of the form
![$\displaystyle \psi(x,t) = e^{-\imath E t/\hbar} \psi(x) .$](img323.png) |
(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
,
![$\displaystyle H \psi = E \psi$](img324.png) |
(6.7) |
or with 6.2
![$\displaystyle - \frac{\hbar^2}{2 m^{*}} \psi'' + V(x) \psi = E \psi .$](img325.png) |
(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
![$\displaystyle \langle \psi \vert \phi \rangle = \int \psi^{\star}(x) \phi(x) dx$](img327.png) |
(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
![$\displaystyle \psi(x) = \langle x \vert \psi \rangle \quad \phi(p) = \langle p \vert \phi \rangle$](img331.png) |
(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
![$\displaystyle \bar A = \langle \psi \vert A \vert \psi \rangle = \langle \psi \vert A \psi \rangle = \langle A \psi \vert \psi \rangle .$](img336.png) |
(6.11) |
Note that for non self-adjoint operators
the notation
is ambiguous.
Previous: 6.1.1 Wave Mechanics
Up: 6.1.1 Wave Mechanics
Next: 6.1.1.2 Von Neumann Equation
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