# The ground state of a time-reversal invariant system must has zero momentum?

• wdlang
In summary, the statement that the ground state of a time reversal invariant Hamiltonian must have zero momentum is not always true. The ground state may spontaneously break time reversal symmetry, resulting in a degenerate ground state. This can be seen in examples such as a ferromagnet at zero temperature or a Hamiltonian describing rotons in He-4. Generally, the momentum operator does not commute with the Hamiltonian, leading to an uncertain momentum in the ground state. This is explained in the Schrodinger time independent equation and can be seen in the RMP paper by I. Bloch et al. (RMP 80, 885 (2008), page 905, paragraph under eq. 65).
wdlang
if the ground state is non-degenearate, this is easily understood

But what if the ground state is non-degenerate?

If you are arguing that the ground state is degenerate with eigenmomenta p and -p, I would argue this is not a state of zero momenta (even though the expectation value of the momentum operator is 0): if you measure p, you never get 0.

Hi wdlang,

The statement that the ground state of a time reversal invariant Hamiltonian must have zero momentum is simply not true.

The ground state may spontaneously break time reversal symmetry. However, if it does, then time reversal symmetry implies the existence of another perfectly good time reversed ground state. Thus the ground state is degenerate in this case.

I can give a simpler example than momentum that may help. Consider a ferromagnet at zero temperature. The ground state has all the spins aligned, but there is an equally good ground state with all spins reversed. The ground state is in fact degenerate.

For the case of momentum, here is a simple example that works. Consider the Hamiltonian H = E0 + (|p| - p0)^2/2m with E0, p0, and m constants . Such a Hamiltonian roughly describes rotons in He-4. The ground states of this hamiltonian have non-zero momentum and are degenerate.

Hope this helps.

Physics Monkey said:
Hi wdlang,

The statement that the ground state of a time reversal invariant Hamiltonian must have zero momentum is simply not true.

The ground state may spontaneously break time reversal symmetry. However, if it does, then time reversal symmetry implies the existence of another perfectly good time reversed ground state. Thus the ground state is degenerate in this case.

I can give a simpler example than momentum that may help. Consider a ferromagnet at zero temperature. The ground state has all the spins aligned, but there is an equally good ground state with all spins reversed. The ground state is in fact degenerate.

For the case of momentum, here is a simple example that works. Consider the Hamiltonian H = E0 + (|p| - p0)^2/2m with E0, p0, and m constants . Such a Hamiltonian roughly describes rotons in He-4. The ground states of this hamiltonian have non-zero momentum and are degenerate.

Hope this helps.

thanks a lot!

When you say "ground state", I assume you mean ground state of the Hamiltonian, i.e. the energy ground state.

Generally speaking, because of the position dependent potential energy term in the Hamiltonian, the momentum operator does not commute with the Hamiltonian. In any energy eigenstate the momentum is uncertain; there is no ground state momentum. Thus, if we measure the momentum in the ground state, there is an entire eigenvalue spectrum of possible results, but there is no unique value .

The Hamiltonian eigenvalue equation is the Schrodinger time independent equation. (time independent??) So, I am confused! Could you please give me a reference where you saw this? Thank you.

Are you referring to a time-dependent Hamiltonian?

eaglelake said:
When you say "ground state", I assume you mean ground state of the Hamiltonian, i.e. the energy ground state.

Generally speaking, because of the position dependent potential energy term in the Hamiltonian, the momentum operator does not commute with the Hamiltonian. In any energy eigenstate the momentum is uncertain; there is no ground state momentum. Thus, if we measure the momentum in the ground state, there is an entire eigenvalue spectrum of possible results, but there is no unique value .

The Hamiltonian eigenvalue equation is the Schrodinger time independent equation. (time independent??) So, I am confused! Could you please give me a reference where you saw this? Thank you.

i saw this statement in the RMP paper by I. Bloch et al.

RMP 80, 885 (2008)

on page 905, the paragraph under eq. 65.

## 1. What is the ground state of a time-reversal invariant system?

The ground state of a time-reversal invariant system is the state of lowest energy, where all particles are in their lowest possible energy levels. This state is considered to be the most stable and is often used as a reference point for comparing other states in the system.

## 2. How is the ground state of a time-reversal invariant system different from other states?

The ground state of a time-reversal invariant system has the lowest energy and the most stable configuration of particles. Other states may have higher energy levels and different configurations of particles.

## 3. Why must the ground state of a time-reversal invariant system have zero momentum?

This is because time-reversal invariance, which is a fundamental symmetry in physics, dictates that the ground state must be unchanged when time is reversed. Since momentum is a vector quantity that changes direction when time is reversed, the ground state must have zero momentum in order to remain unchanged.

## 4. How does the ground state's zero momentum affect the behavior of a time-reversal invariant system?

The zero momentum of the ground state ensures that the system is in a balanced, stable state. This means that any perturbations or changes to the system will not cause it to lose its stability and will return to its original state when the perturbation is removed.

## 5. Are there any exceptions to the rule that the ground state must have zero momentum in a time-reversal invariant system?

In certain systems, such as systems with long-range interactions or ones that exhibit spontaneous symmetry breaking, the ground state may have non-zero momentum. However, these are considered to be special cases and the principle of zero momentum in the ground state still applies in the majority of time-reversal invariant systems.

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