How do you find the probabilities for an anharmonic quantum oscillator state?

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The discussion centers on calculating the probabilities of finding an anharmonic quantum oscillator in various states after a perturbation. The initial state at time t=0 is given as |Ψ(t)⟩ = |1⟩, with the perturbation defined as V(x) = αx³. The corrected state includes contributions from |0⟩, |2⟩, and |4⟩, represented by coefficients c_k, which are derived from the perturbation. There is confusion regarding the normalization of the state and the probabilities, particularly for P_1, which should not equal 1. The final clarification emphasizes that the probabilities should be calculated as P_k = |c_k|²/⟨Ψ|Ψ⟩ to ensure proper normalization.
damarkk
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Homework Statement
Find the probability for all possible unperturbed states ##| n\rangle## for an harmonic oscillator with perturbation ##V(x)=\alpha x^3##
Relevant Equations
##x##, ##p##.
I have one tremendous doubt about it.


On ##t=0## the state of the oscillator is ##| \Psi (t) \rangle = | 1 \rangle ##. The perturbation is ##V(x)=\alpha x^3 = \alpha (\frac{\hbar}{2m\omega})^{3/2} (a+a^{\dagger})^3 = \gamma (a^3+3Na+3Na^{\dagger} + 3a + (a^{\dagger})^3)##.

The only possible other states are ##|0 \rangle##, ##| 2\rangle##, ##|4\rangle##. The state corrected on the first order is:

##|\Psi \rangle = |1 \rangle + c_0|0 \rangle + c_2|2 \rangle + c_4|4\rangle##

where ##c_k = \langle k| V(x) | 1 \rangle ##.


What are the probability to find the oscillator in a generic state ##|n \rangle##?


My answer is ##P_k = |c_k|^2##, but for k=1 we have ##P_1 = 1##. This is not possible I think. On the other hand, I suppose that ##P_1 = 1-P_0 - P_2- P_4 = 1-|c_0|^2-|c_2|^2-|c_4|^2##, but how can I show this? The coefficient of ##| 1\rangle## is 1. This is my question.
 
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damarkk said:
Homework Statement: Find the probability for all possible unperturbed states ##| n\rangle## for an harmonic oscillator with perturbation ##V(x)=\alpha x^3##
Relevant Equations: ##x##, ##p##.

On ##t=0## the state of the oscillator is ##| \Psi (t) \rangle = | 1 \rangle ##.
How do you know the state is ##|1\rangle## at ##t = 0##? This was not given in the homework statement as given. Always provide the full homework statement.

I don't understand "##x##, ##p##" for your Relevant Equations.
damarkk said:
The perturbation is ##V(x)=\alpha x^3 = \alpha (\frac{\hbar}{2m\omega})^{3/2} (a+a^{\dagger})^3 = \gamma (a^3+3Na+3Na^{\dagger} + 3a + (a^{\dagger})^3)##.

The only possible other states are ##|0 \rangle##, ##| 2\rangle##, ##|4\rangle##. The state corrected on the first order is:

##|\Psi \rangle = |1 \rangle + c_0|0 \rangle + c_2|2 \rangle + c_4|4\rangle##

where ##c_k = \langle k| V(x) | 1 \rangle ##.
Your expression for ##c_k## doesn't have the correct dimensions. Did you leave out something?

damarkk said:
What are the probability to find the oscillator in a generic state ##|n \rangle##?


My answer is ##P_k = |c_k|^2##, but for k=1 we have ##P_1 = 1##. This is not possible I think. On the other hand, I suppose that ##P_1 = 1-P_0 - P_2- P_4 = 1-|c_0|^2-|c_2|^2-|c_4|^2##, but how can I show this? The coefficient of ##| 1\rangle## is 1. This is my question.
From ##|\Psi \rangle = |1 \rangle + c_0|0 \rangle + c_2|2 \rangle + c_4|4\rangle##, you can see that ##|\Psi \rangle## is not normalized.

That's ok. The probability ##P_k## is then ##|c_k|^2/\langle \Psi |\Psi \rangle##.
 
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