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To conclude, we formulate our first axiom of quantum mechanics:
Axiom 1: A quantum mechanical system is described by a vector
of a Hilbert space
. The time evolution of
is determined by
the Schrödinger equation
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(174) |
The Hamilton operator
is an operator corresponding to the total
energy of the system. In the case of a single particle with mass
moving in the configuration space
under a potential
,
the wave function
(square
integrable functions) obeys
![$\displaystyle i\hbar\frac{\partial}{\partial t} \Psi({\bf x},t)
=\left[-\frac{\hbar^2\Delta}{2m} + V({\bf x}) \right]\Psi({\bf x},t).$](img125.png) |
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(175) |
is the probability for the particle to be in the (infinitesimal small) volume
around
at time t.
The solutions of the stationary Schrödinger equation at fixed energy,
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(176) |
are called stationary states, the possible energies
eigenenergies.
Note that the form of the Hamilton operator not necessarily has to be as in
(3.10), the Hamiltonian for a single particle defined
over the space
. We will later encounter, for example, Hamiltonians that describe the sites of a finite lattice
and have the form of a
matrix.
Next: Math: Completeness, Dirac notation
Up: Axioms of Quantum Mechanics
Previous: The Hilbert space
Contents
Tobias Brandes
2004-02-04