Can an Electron Be a Frame of Reference in Quantum Mechanics?

[orthogonal]

How does the concept of frame of reference apply to quantum mechanics? Classically something can be a frame of reference as long as it is not accelerating. I often picture in my head an atom with the nucleus fixed and the electron cloud surrounding it. Would it be possible to describe an atom from the perspective of an electron? Is that a valid frame of reference? Something tells me that it isn't but can't come up with a good reason why.

 

[mfb]

Would it be possible to describe an atom from the perspective of an electron? Is that a valid frame of reference?

It is not an inertial frame, so you get all sorts of weird physics equations, but I am sure it is possible.

 

[Nugatory]

Classically something can be a frame of reference as long as it is not accelerating.

Strictly speaking, a frame of reference is a convention for attaching coordinates to points in spacetime, and when you say something can be a frame of reference, you're really saying that there exists a set of coordinates in which that something is at rest. Phrased that way, you don't have a frame of reference question, you have a question about how effectively the electron can be described in terms of points in spacetime.

 

[orthogonal]

you don't have a frame of reference question, you have a question about how effectively the electron can be described in terms of points in spacetime.

Thanks for helping me think through my question. Is this similar to how we determine the at rest mass of an electron? As far as I know an electron is never actually at rest.

 

[Bill_K]

"The reference frame of the electron" is meaningful in classical mechanics, but not in quantum mechanics, because the electron does not have a well-defined trajectory.

 

[tom.stoer]

"The reference frame of the electron" is meaningful in classical mechanics, but not in quantum mechanics, because the electron does not have a well-defined trajectory.

Yes and no.

Suppose you have a classical Hamiltonian function H(q1, q2, p1, p2) for a two-particle system like an electron and a proton. Then usually one fixes the c.o.m. frame classically and introduces the relative momentum p, the corresponding variable x=x2-x1 and the reduced mass m1*m2/(m1+m2). This results in an effective one-particle Hamiltonian h(x,p); usually on choses the c.o.m. frame with P=0.

But there is an alternative description where the full two-particle system is quantized. Then one can apply a unitary transformation, i.e. an operator U which implements a transformation that results in new operators q1', q2', ... and a new Hamiltonian H'. This new Hamiltonian looks like the reduced one-particle Hamiltonian h(p,x) depending on the new variables plus a free Hamiltonian P²/2M with the c.o.m. momentum P and the total mass M=m1+m2. This second term corresponds to the wave function of a free particle with mass M and momentum P. The whole solution is a product of some ψ for x and p and a plane wave (free particle) for P.

In that sense the classical canonical transformation between different frames of reference corresponds to a quantum mechanical unitary transformation acting on operators and states.

 

[Bill_K]

Sure! By a change of coordinates you can either solve the Hydrogen atom in center-of-mass coordinates, or in the original coordinates x1, x2 of the individual particles. But whichever way you do it, the particles move in those coordinates.

H has a kinetic energy term ħ²/2m₁∇₁² + ħ²/2m₂∇₂². And each particle has an orbital angular momentum, and so on. Using the electron's coordinate x₁ does not mean you are "describing an atom from the perspective of an electron".