Stupid question about superposition of quantum states

[Robsta]

Homework Statement

A quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates |n with eigenvalues E_n = (n + 1/2) ħω. Initially, the oscillator is in the state (|0> + |1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the first time for which the time-evolved state is orthogonal to the initial one.

The Attempt at a Solution

I know how to evolve the states in time using the exponential solution to the TDSE.

|ψ(t)> = (e^-iωt/2|0> + e^-3iωt/2|1>)/√2.

This is fine. now I want the probability of finding the system in state |0> at time t. I bra through with <0|
<0|ψ(t)> = (e^-iωt/2<0|0> + e^-3iωt/2<0|1>)/√2.

<0|1> are orthogonal states so their inner product dissapears. <0|0> = 1

<0|ψ(t)> = e^-iωt/2/√2

The probability of finding the state is |<0|ψ(t)>|²

P(|0>) = 1/2

Now here's the problem. When I take the mod square, the time dependence disappears, implying that the probability is constant for all time. This isn't true, at least I don't think it's true. Isn't the probability supposed to oscillate? I've forgotten how this oscillation comes about in bra-ket notation.

 

[nrqed]

Homework Statement

A quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates |n with eigenvalues E_n = (n + 1/2) ħω. Initially, the oscillator is in the state (|0> + |1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the first time for which the time-evolved state is orthogonal to the initial one.

The Attempt at a Solution

I know how to evolve the states in time using the exponential solution to the TDSE.

|ψ(t)> = (e^-iωt/2|0> + e^-3iωt/2|1>)/√2.

This is fine. now I want the probability of finding the system in state |0> at time t. I bra through with <0|
<0|ψ(t)> = (e^-iωt/2<0|0> + e^-3iωt/2<0|1>)/√2.

<0|1> are orthogonal states so their inner product dissapears. <0|0> = 1

<0|ψ(t)> = e^-iωt/2/√2

The probability of finding the state is |<0|ψ(t)>|²

P(|0>) = 1/2

Now here's the problem. When I take the mod square, the time dependence disappears, implying that the probability is constant for all time. This isn't true, at least I don't think it's true. Isn't the probability supposed to oscillate? I've forgotten how this oscillation comes about in bra-ket notation.

There is no problem with your calculation.

However, this is not what they are asking: they are asking the first time at which the state is orthogonal to the initial one so you need \Bigl| \langle \psi(0) | \psi(t) \rangle \Bigr|^2