Difference between stationary/non-stationary quantum states

[acdurbin953]

Homework Statement

I apologize, this is not really a homework problem. I have an exam coming up, and I need to be able to explain the difference between a stationary/non-stationary quantum state in a qualitative way, and in what cases these states have time dependent probabilities. I am hoping someone can correct my understanding if it is wrong. Thank you!

Homework Equations

The Attempt at a Solution

A stationary state is any quantum state which consists of only one eigenstate of the Hamiltonian H. For example, if a spin 1/2 system in the z basis with a magnetic field in the z-direction, a stationary state we may work with is |ψ(0)> = a|+>. In the same system, a non-stationary state would be |ψ(0)> = a|+> + b|->. The important distinction is that stationary states are composed of one energy eigenstate of the H, and non-stationary states are a superposition of n energy eigenstates of H (for a spin 1/2 system, this would only be up to n=2).

Probabilities of any state are time independent if:

• The state we are measuring the probability in is stationary OR
• We are measuring the probability in a basis that commutes with the basis of the Hamiltonian.

Probabilities of any state are time dependent if:

• The state is non-stationary and we are measuring the probability in a basis that does not commute with the basis of the Hamiltonian.

Also sorry if this is a dumb question, but is it possible to have a state that is not made up of energy eigenvalues/eigenstates of the Hamiltonian? I don't think you can.

 

[DrClaude]

Also sorry if this is a dumb question, but is it possible to have a state that is not made up of energy eigenvalues/eigenstates of the Hamiltonian?

No. The eigenstates of the Hamiltonian form a complete basis, as do the eigenstates of any observable. Therefore, it is always possible to write any state as a linear combination of those eigenstates.

 

[blue_leaf77]

When one talks about stationary states, it's actually more about the time dependency of the observables. If a system is known to be in a stationary state, then the expectation value of any observable quantity measured on this system will be time independent. If on the other hand the state is non-stationary state, any observables which do not commute with the Hamiltonian will have time-dependent expectation value.

The important distinction is that stationary states are composed of one energy eigenstate of the H

Not always, consider this superposition state in a hydrogen atom $|\psi\rangle = \frac{\sqrt{3}}{2}|u_{211}\rangle+\frac{1}{2}|u_{200}\rangle$. This state is a stationary state.

As for the time-dependency of the probabilities, it makes more sense to talk about the (non)stationary states as a basis, not as the state of the system. It's just a matter of which basis you want to use to expand the state of a given system.

 

[acdurbin953]

When one talks about stationary states, it's actually more about the time dependency of the observables. If a system is known to be in a stationary state, then the expectation value of any observable quantity measured on this system will be time independent. If on the other hand the state is non-stationary state, any observables which do not commute with the Hamiltonian will have time-dependent expectation value.

Not always, consider this superposition state in a hydrogen atom $|\psi\rangle = \frac{\sqrt{3}}{2}|u_{211}\rangle+\frac{1}{2}|u_{200}\rangle$. This state is a stationary state.

As for the time-dependency of the probabilities, it makes more sense to talk about the (non)stationary states as a basis, not as the state of the system. It's just a matter of which basis you want to use to expand the state of a given system. How can it be stationary?

The bold text makes sense to me, however I'm confused now about the hydrogen atom state. Is the state you wrote a spin 1/2 state? I think we are only dealing with spin 1/2 states, and we have yet to have any homework/practice problems where a superposition state was stationary and any observable quantity measured was time independent.

 

[blue_leaf77]

Is the state you wrote a spin 1/2 state?

No, they are the eigenfunctions of hydrogen atom Hamiltonian $u_{nlm}$.

I think we are only dealing with spin 1/2 states

Why do you have to specialize the discussion on the spin 1/2 states only. The concept of stationary states applies to all kinds of Hamiltonian.

we have yet to have any homework/practice problems where a superposition state was stationary

Stationary states are equivalent to the eigenstates of the system's Hamiltonian. Keeping this in mind, a superposition state can be a stationary state if the superposing states all have the same energy. For example of hydrogen atom, the energy is a function of the principal quantum number, $n$, only. Therefore, eigenstates $|u_{nlm}\rangle$ with different $l$ and $m$ but the same $n$ can superpose to form a stationary state.

 

[acdurbin953]

No, they are the eigenfunctions of hydrogen atom Hamiltonian $u_{nlm}$.

Why do you have to specialize the discussion on the spin 1/2 states only. The concept of stationary states applies to all kinds of Hamiltonian.

Stationary states are equivalent to the eigenstates of the system's Hamiltonian. Keeping this in mind, a superposition state can be a stationary state if the superposing states all have the same energy. For example of hydrogen atom, the energy is a function of the principal quantum number, $n$, only. Therefore, eigenstates $|u_{nlm}\rangle$ with different $l$ and $m$ but the same $n$ can superpose to form a stationary state.

Hmm, I should have prefaced my question - the class I am in is an intro course, and our professor told us we'd only be working with either spin 1/2 or spin 1 systems. I know that the concept applies to all kinds of Hamiltonian operators. For this exam it was specified that all systems would be spin 1/2, so I was just looking to make sure I understood that type of system.

 

[blue_leaf77]

Hmm, I should have prefaced my question - the class I am in is an intro course, and our professor told us we'd only be working with either spin 1/2 or spin 1 systems. I know that the concept applies to all kinds of Hamiltonian operators. For this exam it was specified that all systems would be spin 1/2, so I was just looking to make sure I understood that type of system.

Every system which involves spin-1/2 particles are called spin-1/2 system. A hydrogen atom is also a spin-1/2 system because the electron (and proton) is a spin-1/2 particle, however in the nonrelativistic limit, the spin makes no effect at all to the energy levels. This means, for a given spatial wavefunction $u_{nlm}(r,\theta,\phi)$, both spin up and down have the same energy. If the particle (of whatever spin value) is in a region of uniform magnetic field like what you used as an example above, the energy is not degenerate - it depends on the orientation of the spin. In this case, no superposition state can be a stationary state.

In general, you have to pay attention to what kind of Hamiltonian the particle is subject to. Although your prof told you that he will only consider spin-1/2 or 1 particle, that doesn't mean the system will always be that of a uniform magnetic field.