Talk:Canonical commutation relation

From Wikipedia, the free encyclopedia

WikiProject Physics This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
Start This article has been rated as Start-Class on the assessment scale.
??? This article has not yet received an importance rating within physics.

Help with this template This article has been rated but has no comments. If appropriate, please review the article and leave comments here to identify the strengths and weaknesses of the article and what work it will need.

[edit] Question

Anyone read Dirac's book on QM? I'm missing something on this derivation. -ub3rm4th
Quoted (section 22, page 93):

(\frac{\partial}{\partial q_r})^* \left.\psi\right\rangle^* = e^{-i\gamma}\frac{\partial}{\partial q_r} \left.\psi\right\rangle = e^{-i\gamma}\frac{\partial}{\partial q_r} e^{i\gamma}\left.\psi\right\rangle^*

showing that

(\frac{\partial}{\partial q_r})^* = e^{-i\gamma}\frac{\partial}{\partial q_r} e^{i\gamma}

or, with the help of \frac{\partial}{\partial q_r}f - f\frac{\partial}{\partial q_r} = \frac{\partial f}{\partial q_r},

(\frac{\partial}{\partial q_r})^* = \frac{\partial}{\partial q_r} + i\frac{\partial\gamma}{\partial q_r}

[edit] Question, more urgent!

\frac{\partial}{\partial q_r} \left. \psi\right\rangle = 0

Why?

[edit] Gauge invariant?

"The non-relativistic Hamiltonian for a quantized charged particle of mass m in a classical electromagnetic field is

H=\frac{1}{2m} \left(p-\frac{eA}{c}\right)^2 +e\phi

where A is the three-vector potential and φ is the scalar potential. This form of the Hamiltonian, as well as the Schroedinger equation H\psi=i\hbar \partial\psi/\partial t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

A\to A^\prime=A+\nabla \Lambda
\phi\to \phi^\prime-\frac{1}{c} \frac{\partial \Lambda}{\partial t}
\psi\to\psi^\prime=U\psi
H\to H^\prime= U HU^\dagger

where

U=\exp \left( \frac{ie\Lambda}{\hbar c}\right)

and Λ = Λ(x,t) is the gauge function."


It is not true that this H is invariant under a gauge transformation. The kinetic energy term is gauge invariant but the potential energy term is not so H' becomes H - (e/c)dLambda/dt (making allowance for formatting). However the Schroedinger equation, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation. Xxanthippe 04:02, 11 June 2007 (UTC)