- #1

Robsta

- 88

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## 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.