Galilei Transformation Free Schrödinger Equation

[VVS]

Homework Statement

I am supposed to show that the free Schrödinger Equation is NOT kovariant under Galilei Transformation.

Homework Equations

We learned in Lectures that the Galilei Transformation can be written as:
$\vec{x'}=\hat{R}\vec{x}-\vec{a}-\vec{v}t$ (1) or equivalently:
$\vec{x}=\hat{R^{-1}}\left(\vec{x'}+\vec{a}+\vec{v}t\right)$ (2)

3. The Attempt at a Solution
I have no problem with evaluating the LHS of the time dependent Schrödinger equation.
However I am not sure if the second part of this is needed or not correct.
$\frac{\partial}{\partial x_i}=\sum_k \frac{\partial x'_k}{\partial x_i}\frac{\partial}{\partial x'_k}+\frac{\partial t}{\partial x_i}\frac{\partial}{\partial t}$

 

[vanhees71]

This is a very strange question, because non-relativistic quantum theory is of course Galilei covariant. I don't understand, why you want to derive $t$ with respect to $x_i$. The Galilei trafo for time is $t'=t-b$. Concerning time there are only time translations in non-relativistic physics (Newton's idea of an absolute time).

You'll see that the wave function transforms non-trivially under Galilei boosts, which is an (Abelian) subgroup of the full Galilei transformations you wrote down. It's defined by $\hat{R}=\mathbb{1}_3$, $\vec{a}=0$, i.e., it's the change from one inertial frame to another one moving with velocity $\vec{v}$ with respect to the former.

But then remember, what's observable in quantum theory. Then you'll see that observable quantities do not change under Galilei boosts (and also not under the full Galilei transformation).

 

[dextercioby]

Indeed, the Schrödinger equation written like this:

$$\frac{\partial\psi(\vec{x},t)}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2 \psi(x,t)$$

is Galilei invariant, at least according to page 132 of Fushchich and Nikitin's book: https://www.amazon.com/dp/0898640695/?tag=pfamazon01-20

 

[VVS]

Hey, thanks for your attempts to help me. I solved the problem. The Schrödinger equation is not covariant under Galileitransformations, but can be made so by multiplying the wavefunction with a plane wave function.

 

[vanhees71]

Again: Non-relativistic quantum mechanics IS COVARIANT under Galileo transformations. Otherwise it wouldn't make sense!

Take one free particle as the most simple case. Then the symmetry is realized as
$$t'=t, \quad \vec{x}'=\vec{x}-\vec{v} t, \quad \vec{p}'=\vec{p}-m \vec{v}, \psi'(t',\vec{x}')=\exp(\mathrm{i} \Phi) \psi(t,\vec{x})=\exp(\mathrm{i} \Phi) \psi(t',\vec{x}'+\vec{v} t').$$
Now we have to check that we can find a real phase $\Phi(t',\vec{x}')$ such that $\psi'$ fulfills the Schrödinger equation in the new variables, if $\psi(t,\vec{x})$ fulfills it in the old variables. We have
$$\mathrm{i} \hbar \partial_{t'} \psi'=\mathrm{i} \hbar \exp(\mathrm{i} \Phi) \left [\mathrm{i} \partial_{t'} \Phi(t',\vec{x}') + \vec{v} \cdot \vec{\nabla} \psi(t,\vec{x})|_{t=t',\vec{x}=\vec{x}'+\vec{v} t'} \right ].$$
Further we find
$$\Delta' \psi'(t',\vec{x}') = \left [\mathrm{i} \Delta' \Phi(t',x') \psi(t',\vec{x}'+\vec{v} t') +2 \mathrm{i} \vec{\nabla} ' \Phi(t',\vec{x}') \cdot \vec{\nabla}' \psi(t',\vec{x}'+\vec{v} t') + \Delta ' \psi(t',\vec{x}'+\vec{v} t') + (\vec{\nabla}' \Phi)^2 \psi(t',\vec{x}'+\vec{v} t') \right ]\exp(\mathrm{i} \Phi)$$
Now you demand that
$$\mathrm{i} \hbar \partial_{t'} \psi(t',\vec{x}') = -\frac{\hbar^2}{2m} \Delta' \psi(t',\vec{x}'),$$
provided that
$$\mathrm{i} \hbar \partial_{t} \psi(t,\vec{x}) = -\frac{\hbar^2}{2m} \Delta \psi(t,\vec{x}).$$
Comparing the coefficients in front of $\vec{\nabla} \psi$ and $\psi$, you get the system of PDEs for the phase factors:
$$\vec{\nabla}' \Phi(t',\vec{x}') = -\frac{m}{\hbar} \vec{v}, \quad \partial_{t'} \Phi(t',\vec{x}') = \frac{\hbar}{2m} \mathrm{i} \Delta' \Phi(t',\vec{x}') -\frac{\hbar}{2m} [\vec{\nabla}' \phi(t',\vec{x}')]^2.$$
From the first equation you get
$$\Phi(t',\vec{x}')=-\frac{m}{\hbar} \vec{v} \cdot \vec{x} + \Phi_1(t')$$
and from the 2nd
$$\dot{\Phi}_1(t')=-\frac{\hbar}{2m} \vec{v}^2 \; \Rightarrow \; \Phi=-\frac{m}{\hbar} \vec{v} \cdot \vec{x}' - \frac{\hbar}{2m} \vec{v}^2 t' + \phi_0.$$
Thus the transformation law for the Schrödinger wave function under Galileo boosts reads
$$\psi'(t',x')=\exp\left [\frac{m}{\hbar} \vec{v} \cdot \vec{x}' - \frac{\hbar}{2m} \vec{v}^2 t' + \phi_0 \right ] \psi(t',\vec{x}'+\vec{v} t').$$
This means that up to a phase factor the wave function is a scalar under Galileo boosts and thus quantum theory is covariant under Galileo boosts, because the measurable quantities are all invariant under the Galileo transformation.

The Galileo group is realized as a unitary ray representation. There is no sensible realization of the Galileo group as a unitary transformation. The mass is a non-trivial central charge of the Galileo algebra, and setting it to 0 does not lead to useful quantum dynamics. Massless particles make no sense in non-relativistic physics at all. For a deeper understanding, see

Inönü, E., Wigner, E. P.: Representations of the Galilei group, Il Nuovo Cimento 9(8), 705–718, 1952
http://dx.doi.org/10.1007/BF02782239

Ballentine, Leslie E.: Quantum Mechanics, World Scientific, 1998

 

[VVS]

Thanks for the derivation.
Then there must be some confusion about the exact definition of covariance. Because our professor asked us to prove that the SCHRÖDINGER EQUATION is not covariant under Galilei Transformations. And you can actually show that if you take the Schrödinger equation in the moving system and perform the Galilei-Transformation then you yield the Schrödinger equation in the resting system, but with an additive term.
Then he asked us in a follow up question to determine the factor by which we have to multiply the wave function in the resting system to make it Galilei covariant.

 

[vanhees71]

Well, it may be a question of definition, but what does "covariance" or "invariance" mean? It means that the physically relevant quantities in a theory are independent under a symmetry transformation. In quantum theory the wave function is not directly observable but its modulus squared is the probability distribution for the position of the particle and thus a measurable quantity. This probability distribution should be Lorentz covariant, because otherwise quantum theory would not be consistent with the space-time structure of Newtonian mechanics and wouldn't make sense within a "Newtonian" quantum theory.

Together with the general structure of quantum theory, this implies that the wave function (or more generally the Hilbert-space vectors representing (pure) states) are only defined up to a phase factor, and a symmetry transformation is only determined up to phase factors. Thus, the true representatives of the pure states are not the (normalized) Hilbert-space vectors but the rays in Hilbert space, i.e., each normalized vector can be multiplied by phase factors without changing its physical meaning.

Consequently a symmetry transformation is represented by either a unitary or antiunitary ray representation (Wigner-Bargmann theorem). This is of utmost importance. First, as you have seen in your exercise, the very valid Schrödinger theory (non-relativistic quantum theory) cannot be reformulated in a way that you have a true unitary representation of the Galilei group. Because the Galilei group is a Lie group, i.e., a continuous group, the transformation must be a unitary ray representation. Now you can ask, whether it's possible to find a truly unitary representation of the Galilei group on Hilbert space. Of course one can, but it turns out that it doesn't lead to a physically sensible dynamics.

The second reason for the importance that rays and not vectors represent states is the well-established existence of half-integer spins. For half-integer spinor representations of the rotation group (which is a subgroup of the Galilei group) the rotation around $360^{\circ}$ leads to a phase factor $-1$, but that doesn't matter, because multiplying all vectors by the same phase factor is irrelevant. This of course further implies that there must not be superpositions of states from a half-integer spin and an integer-spin particle, because a rotation around $360^{\circ}$ would lead to a different state, because the former pieces multiyply by -1 but the latter don't change. This shouldn't happen, because a rotation around $360^{\circ}$ means that you've done nothing at all, i.e., it's the identity transformation within the physical Galilei group. Indeed, so far nobody has ever found a superposition of half-integer and ingeger-spin states.

 

[VVS]

Thank you for your comment. I learned not only from doing the exercise but also from your comments.

 

[Valera]

Again: Non-relativistic quantum mechanics IS COVARIANT under Galileo transformations. Otherwise it wouldn't make sense!

Take one free particle as the most simple case. Then the symmetry is realized as
$$t'=t, \quad \vec{x}'=\vec{x}-\vec{v} t, \quad \vec{p}'=\vec{p}-m \vec{v}, \psi'(t',\vec{x}')=\exp(\mathrm{i} \Phi) \psi(t,\vec{x})=\exp(\mathrm{i} \Phi) \psi(t',\vec{x}'+\vec{v} t').$$
Now we have to check that we can find a real phase $\Phi(t',\vec{x}')$ such that $\psi'$ fulfills the Schrödinger equation in the new variables, if $\psi(t,\vec{x})$ fulfills it in the old variables. We have
$$\mathrm{i} \hbar \partial_{t'} \psi'=\mathrm{i} \hbar \exp(\mathrm{i} \Phi) \left [\mathrm{i} \partial_{t'} \Phi(t',\vec{x}') + \vec{v} \cdot \vec{\nabla} \psi(t,\vec{x})|_{t=t',\vec{x}=\vec{x}'+\vec{v} t'} \right ].$$
Further we find
$$\Delta' \psi'(t',\vec{x}') = \left [\mathrm{i} \Delta' \Phi(t',x') \psi(t',\vec{x}'+\vec{v} t') +2 \mathrm{i} \vec{\nabla} ' \Phi(t',\vec{x}') \cdot \vec{\nabla}' \psi(t',\vec{x}'+\vec{v} t') + \Delta ' \psi(t',\vec{x}'+\vec{v} t') + (\vec{\nabla}' \Phi)^2 \psi(t',\vec{x}'+\vec{v} t') \right ]\exp(\mathrm{i} \Phi)$$
Now you demand that
$$\mathrm{i} \hbar \partial_{t'} \psi(t',\vec{x}') = -\frac{\hbar^2}{2m} \Delta' \psi(t',\vec{x}'),$$
provided that
$$\mathrm{i} \hbar \partial_{t} \psi(t,\vec{x}) = -\frac{\hbar^2}{2m} \Delta \psi(t,\vec{x}).$$
Comparing the coefficients in front of $\vec{\nabla} \psi$ and $\psi$, you get the system of PDEs for the phase factors:
$$\vec{\nabla}' \Phi(t',\vec{x}') = -\frac{m}{\hbar} \vec{v}, \quad \partial_{t'} \Phi(t',\vec{x}') = \frac{\hbar}{2m} \mathrm{i} \Delta' \Phi(t',\vec{x}') -\frac{\hbar}{2m} [\vec{\nabla}' \phi(t',\vec{x}')]^2.$$
From the first equation you get
$$\Phi(t',\vec{x}')=-\frac{m}{\hbar} \vec{v} \cdot \vec{x} + \Phi_1(t')$$
and from the 2nd
$$\dot{\Phi}_1(t')=-\frac{\hbar}{2m} \vec{v}^2 \; \Rightarrow \; \Phi=-\frac{m}{\hbar} \vec{v} \cdot \vec{x}' - \frac{\hbar}{2m} \vec{v}^2 t' + \phi_0.$$
Thus the transformation law for the Schrödinger wave function under Galileo boosts reads
$$\psi'(t',x')=\exp\left [\frac{m}{\hbar} \vec{v} \cdot \vec{x}' - \frac{\hbar}{2m} \vec{v}^2 t' + \phi_0 \right ] \psi(t',\vec{x}'+\vec{v} t').$$
This means that up to a phase factor the wave function is a scalar under Galileo boosts and thus quantum theory is covariant under Galileo boosts, because the measurable quantities are all invariant under the Galileo transformation.

The Galileo group is realized as a unitary ray representation. There is no sensible realization of the Galileo group as a unitary transformation. The mass is a non-trivial central charge of the Galileo algebra, and setting it to 0 does not lead to useful quantum dynamics. Massless particles make no sense in non-relativistic physics at all. For a deeper understanding, see

Inönü, E., Wigner, E. P.: Representations of the Galilei group, Il Nuovo Cimento 9(8), 705–718, 1952
http://dx.doi.org/10.1007/BF02782239

Ballentine, Leslie E.: Quantum Mechanics, World Scientific, 1998

Hi @vanhees71, sorry to bump an old post. Thanks for presenting that nice derivation here, there's just one step in it that's bugging me: when you match up coefficients for $\psi$ and $\nabla\psi$, it seems like the $\Delta^{'}\psi$ term is ignored when comparing coefficients for the latter (indeed, this introduces a strange $+\nabla$ to the mix if accounted for).