Prove that TDSE is invariant under Galilean Transformation.

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• Viraj Daniel Dsouza
In summary, the canonical commutation relation between wavefunctions doesn't change under Galilean transformations, but the commutator at any time also remains the same.

Viraj Daniel Dsouza

I want the proof for a general wavefunction Ψ(x,t).

The Schrödinger equation isn't invariant under Galilean transformations. It's invariant under a central extension of the universal cover of the Galilean group. Specifically, this means that you cannot just apply a Galilean transformation to the wave-function, but you need to add a (mass and time dependent) phase factor in order to transform a solution into another solution. In contrast to the Lorentz group, the phase factor cannot be made to vanish in case of the Galilean group.

Viraj Daniel Dsouza said:
I want the proof for a general wavefunction Ψ(x,t).

A proof of what?

PeterDonis said:
A proof of what?

I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.

Viraj Daniel Dsouza said:
I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.

See chapter 3 Ballentine - QM - A Modern Development - he does the converse - proves the Schrodinger equation from the Galilean principle of relativity. Specifically he assumes the probabilities of an observation are invariant.

BTW I think Rubi is right although I haven't gone deep into it.

Had a quick scan through Ballentine and Rubi is correct. Here is a paper I found on it - note the phase change mentioned by Rubi.

It says it is invariant - but with that phase change things are a bit murkier. Interesting question - before going into it I would have said yes it is - however its a bit more subtle.

Thanks
Bill

Last edited:
Viraj Daniel Dsouza
Could this be somehow deduced from the fact that the canonical commutation relation ##[\hat{x},\hat{p}]=i\hbar## doesn't change in the transformation

##\hat{x} \rightarrow \hat{x}+vt##,
##\hat{p} \rightarrow \hat{p} + mv##

with ##m## and ##v## real numbers and ##t## the time?

On the other hand, the commutator at any time ##t## will also remain the same if the ##vt## is replaced by ##\beta t^2## or something else that is nonlinear in ##t##.

1. What is the TDSE and what does it stand for?

The TDSE stands for Time-Dependent Schrödinger Equation. It is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system.

2. What is a Galilean Transformation?

A Galilean Transformation is a mathematical transformation that describes the relationship between the coordinates and velocities of a system in different frames of reference. It is commonly used in classical mechanics to understand the motion of objects.

3. How do you prove that TDSE is invariant under Galilean Transformation?

To prove that TDSE is invariant under Galilean Transformation, we need to show that the form of the equation remains the same in different frames of reference. This can be done by applying the transformation to the TDSE and showing that it yields the same equation.

4. What does it mean for TDSE to be invariant under Galilean Transformation?

If the TDSE is invariant under Galilean Transformation, it means that the equation remains unchanged in different frames of reference. This is important because it shows that the fundamental laws of quantum mechanics are not affected by changes in the observer's perspective.

5. Why is it important for TDSE to be invariant under Galilean Transformation?

It is important for TDSE to be invariant under Galilean Transformation because it allows us to apply the principles of quantum mechanics to systems that are in motion. This is essential for understanding the behavior of particles in different frames of reference and has practical applications in fields such as atomic and molecular physics.

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