QM- Probability of finding a system in a superposition of energy eigenstates.

[L-x]

Homework Statement

A system at time t = 0 is in the state |ψ> = a|E1> + b|E2>, where |E1> and |E2>
are (normalised) energy eigenstates with two different energies E1 and E2, and a, b
are real numbers. Write down the state |ψ, t> for the system at time t. What are the
probabilities at time t to find the system in the states |E1>, |E2> and (|E1>+|E2>)/√2?

Homework Equations

TDSE, TISE

The Attempt at a Solution

I believe I've correctly found that the probability to find the system in ether |E1> or |E2> is |a|^2 or |b|^2 respectively independent of time, as solving the TDSE for |ψ> shows that only the phase of a(t), b(t) changes with time. If this is wrong please tell me so i can post my full working!

My question though, is if I've correctly interpreted "find the system in the states" to mean "measure the energy of the system to be", then isn't the probability of finding the system in a superposition of eigenstates 0, as when we measure the superposition of eigenstates collapses into only a single one, so we must obtain either |E1> or |E2>?

Thanks in advance for your help.

 

[collinsmark]

I believe I've correctly found that the probability to find the system in ether |E1> or |E2> is |a|^2 or |b|^2 respectively independent of time, as solving the TDSE for |ψ> shows that only the phase of a(t), b(t) changes with time. If this is wrong please tell me so i can post my full working!

If I remember my QM correctly, you are correct. (Although I wouldn't say that a and b themselves change with time. a and b are constants [and in this particular problem, real constants]. But there are additional time dependent terms tacked on to each [time-independent] energy eigenstate that do have phases that vary with time. I think that's what you mean.)

The tricky part comes when dealing the expectation values of position and momentum, which are time dependent and oscillate back and forth, given the above superposition of energy eigenstates. But the probability of measuring a particular energy value is not time dependent, as you say, and is a function of a² or b² [and assuming that a² + b² = 1]. So yes, I think you are correct.

My question though, is if I've correctly interpreted "find the system in the states" to mean "measure the energy of the system to be", then isn't the probability of finding the system in a superposition of eigenstates 0, as when we measure the superposition of eigenstates collapses into only a single one, so we must obtain either |E1> or |E2>?

Yes, I also agree with you on that. No matter what you measure, the wave-function will collapse (at least at that instant) to an eigenstate of whatever observable is being measured. After that, the wave-function might/will evolve according to Schrodinger's equation. But at least at the moment of collapse, it's a single eigenstate, and not a superposition thereof.

[Edit: In reality, that last statement becomes more complicated when we consider the precision of the instrumentation doing the measurements, particularly when dealing with continuous observables such as position or momentum. It's also a bit more complicated in situations such as measuring the momentum of a particle in an infinite square well, since the wave-function, represented in momentum space, is not allowed to be a true Dirac delta function (regardless of the instrumentation's precision). But these concepts are beyond the scope of this problem, so I don't even want to discuss them here.]

 

[vela]

My question though, is if I've correctly interpreted "find the system in the states" to mean "measure the energy of the system to be", then isn't the probability of finding the system in a superposition of eigenstates 0, as when we measure the superposition of eigenstates collapses into only a single one, so we must obtain either |E1> or |E2>?

"Find the system in the states" isn't synonymous with "measure the energy of the system to be." Assume there's some observable O that has an eigenstate |ϕ>=(|E1>+|E2>)/√2. What's the probability that you measure O and collapse the system to that state? That's what they're asking you to find.

 

[collinsmark]

"Find the system in the states" isn't synonymous with "measure the energy of the system to be." Assume there's some observable O that has an eigenstate |ϕ>=(|E1>+|E2>)/√2. What's the probability that you measure O and collapse the system to that state? That's what they're asking you to find.

Thanks Vela for chiming in. I trust your opinion on QM. But I don't think I follow your answer on this one.

To me, your description rings of something along the lines: Consider a uniform random variable bounded by [0.00, 1.00). What's the probability that at any given instant in time, the variable will be exactly 0.707106781185...? But since a range is not specified, only a specific value, the probability is 0. It's almost like asking: if a and b are continuous, real numbers along [0, 1], and the only restriction is that a² + b² = 1, what's the probability that a = b?

Allow me to continue in a different light. Suppose there is such an observable O, that has a set of discrete eigenstates. (That way we can ignore the continuum that brought up in the last paragraph). In order for |ϕ>=(|E1>+|E2>)/√2 to have a finite probability, |ϕ>=(|E1>+|E2>)/√2 has to be one of the discrete eigenstates of O. But given the limited information in the problem statement there's no way to determine whether |ϕ>=(|E1>+|E2>)/√2 is one of these discrete eigenstates or anything about what the eigenstates of O would be. Perhaps if the problem statement specified the potential, and specified what the operator O was, it could be solvable. But other than that, there's just not enough information, as far as I can tell.

 

[vela]

I only brought up the hypothetical observable O so one could see it's plausible to ask what the probability it is of finding a system in a specific superposition of states.

Say a system is in the state $\vert \psi \rangle$ and you want to know the probability of finding it in a state $\vert \phi \rangle$. The general procedure is this: You first calculate the amplitude $\langle \phi \vert \psi \rangle$. The probability is then given by the modulus of the amplitude squared.

Another way to look at it is through a change of basis. Your original basis is {|E1>, |E2>}, and the new orthonormal basis is {|B1>, |B2>} where |B1>=(|E1>+|E2>)/√2 and |B2>=(|E1>-|E2>)/√2}. You can find c(t) and d(t) such that |ψ(t)> = c(t)|B1> + d(t)|B2>. Then the probability of finding it in the state |B1> at time t is the usual |c(t)|².

You should get the same answer either way, but the first way is probably simplest for this problem.

 

[collinsmark]

Ah. When you explain it that way, it makes a lot of sense. And you can use the very same method easily find the respective probabilities of |E1> and |E2> too (by using the $\langle \phi \vert \psi \rangle$ approach, that is). Hmph. Well, there ya' go. http://www.websmileys.com/sm/happy/1172.gif

 

[L-x]

Thanks very much for your help collinsmark and vela.