States of a Quantum Harmonic Oscillator
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6. Consider the state of a harmonic oscillator initially (t=0) to be given by |phi >= 5|0 + 12| 1>.
(a) Find the normalized state.
(b) What will be the state of the particle after time t.
(c) Calculate < x > and < p > for this state at time t. Is this classically what you would expect?
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Solution Summary
The solution shows in detail how to normalize the wavefunction and its time evolution and then how to calculate the position and momentum expectation values using the ladder operators. It shows the result complies with classical interpretation. The expert provides the solution in a PDF and a word document.
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If we write the state as a superposition of orthonormal eigenstates then:
(1.1)
Normalization requires that:
(1.2)
And we are given that:
(1.3)
We solve for :
We get:
(1.4)
And:
(1.5)
So the normalized state is:
(1.6)
The energy associated with orthonormal state is:
(1.7)
The time evolution of the eigenstate is:
(1.8)
So in our case the time evolution of the superimposed state is:
(1.9)
The expectation ...
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