- #1
Syrius
- 8
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Hello everybody,
I am currently struggling with a problem that I came across while spending some free time on non-relativistic quantum mechanic problems.
Suppose we have an electron that is describe at time [itex] t_0 = 0 [/itex] by a wave function in position space [itex]\psi(x,y,z)[/itex]. Furthermore, assume that we have a Hamiltonian
[itex] \hat{H}_1 = \frac{1}{2m}\left [ \mathbf{p}+\frac{e}{c} \mathbf{A}_1(\mathbf{r}) \right ]^2[/itex]
with charge e, mass m and speed of light c. Then we can construct another Hamiltonian
[itex] \hat{H}_2 = \frac{1}{2m} \left [\mathbf{p}+\frac{e}{c} \mathbf{A}_2(\mathbf{r}) \right ]^2[/itex]
that can be obtained from [itex] \hat{H}_1[/itex] via a gauge transformation of the vector potential
[itex] \mathbf{A}_2(\mathbf{r}) = \mathbf{A}_1(\mathbf{r}) + ∇ f(\mathbf{r}) [/itex]
with a scalar function f.
The question is now which of these two Hamiltonians do I use to evolve my initial wave function in time? I mean, if I fix the initial wave function and the Hamiltonian it is like fixing a gauge for the initial wave function.
In other words, are there some conditions that the initial wave function has to fulfill that depend on the chosen gauge?
Greetings, Syrius
I am currently struggling with a problem that I came across while spending some free time on non-relativistic quantum mechanic problems.
Suppose we have an electron that is describe at time [itex] t_0 = 0 [/itex] by a wave function in position space [itex]\psi(x,y,z)[/itex]. Furthermore, assume that we have a Hamiltonian
[itex] \hat{H}_1 = \frac{1}{2m}\left [ \mathbf{p}+\frac{e}{c} \mathbf{A}_1(\mathbf{r}) \right ]^2[/itex]
with charge e, mass m and speed of light c. Then we can construct another Hamiltonian
[itex] \hat{H}_2 = \frac{1}{2m} \left [\mathbf{p}+\frac{e}{c} \mathbf{A}_2(\mathbf{r}) \right ]^2[/itex]
that can be obtained from [itex] \hat{H}_1[/itex] via a gauge transformation of the vector potential
[itex] \mathbf{A}_2(\mathbf{r}) = \mathbf{A}_1(\mathbf{r}) + ∇ f(\mathbf{r}) [/itex]
with a scalar function f.
The question is now which of these two Hamiltonians do I use to evolve my initial wave function in time? I mean, if I fix the initial wave function and the Hamiltonian it is like fixing a gauge for the initial wave function.
In other words, are there some conditions that the initial wave function has to fulfill that depend on the chosen gauge?
Greetings, Syrius