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I seem to have two approaches that I've seen and understand, but I can't quite see how they relate.
1. Write a general time evolving state as a superposition of stationary states multiplied by their exp(-iEt/h) factors, and calculate <x>. We find that <x>=Acos(wt+b) as in classical physics (in the limit of large energies, the time evolving probability distribution spikes around this time evolving expectation value, so this works).
2. Look at the probability distribution of a stationary state for large n, and compare it to the probability distribution for a classical harmonic oscillator, and note that they are similar (if you just consider the envelope of the quantum mechanical prediction). However in this method, <x>=0.
So method 1 says that a classical oscillator has to be in a superposition of stationary states, method 2 says it is in a single stationary state. Method 1 is good on the grounds that we retrieve our usual oscillatory result, method 2 is good in that we see the probability distributions match. If you try to cross the methods over and prove the opposite things for each, then method 2 says <x>=0 (not good!), method 1 gives a time evolving probability distribution (hence <x>=f(t)), and so they disagree.
Each approach alone seems to make sense to me, but then they seem to be disagreeing when I compare them which is confusing me. Can anyone help, thanks :)
1. Write a general time evolving state as a superposition of stationary states multiplied by their exp(-iEt/h) factors, and calculate <x>. We find that <x>=Acos(wt+b) as in classical physics (in the limit of large energies, the time evolving probability distribution spikes around this time evolving expectation value, so this works).
2. Look at the probability distribution of a stationary state for large n, and compare it to the probability distribution for a classical harmonic oscillator, and note that they are similar (if you just consider the envelope of the quantum mechanical prediction). However in this method, <x>=0.
So method 1 says that a classical oscillator has to be in a superposition of stationary states, method 2 says it is in a single stationary state. Method 1 is good on the grounds that we retrieve our usual oscillatory result, method 2 is good in that we see the probability distributions match. If you try to cross the methods over and prove the opposite things for each, then method 2 says <x>=0 (not good!), method 1 gives a time evolving probability distribution (hence <x>=f(t)), and so they disagree.
Each approach alone seems to make sense to me, but then they seem to be disagreeing when I compare them which is confusing me. Can anyone help, thanks :)