![]() |
(7.11) |
To regain the Schrödinger equation from the von Neumann equation
we make a separation ansatz for the density :
![]() |
(7.12) |
We get (omitting in the function arguments)
![]() |
(7.13) |
Factoring out and
gives
![]() |
(7.14) |
Division by
separates the equation and we get (with
as separation constant)
If is a solution to Equation 7.15 with separation constant
, then
is a solution to Equation 7.16 with complex conjugate separation constant
and the density matrix
is of the form
corresponding
to a pure state.
We get an additional term
in the separated equation, which
is at first surprising since we are not in the stationary case. However,
the von Neumann equation is invariant under a change of
to
,
while the Schrödinger equation is not.
By separation of the transient von Neumann equation we get
the Schrödinger equation with an additional term
.
This shifts the value of
in the stationary Schrödinger
equation. But varying
the set of solutions
stays the same.
So we can set
without loss of generality.
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