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Hi all
I have the first three states of the harmonic oscillator, and I need to know the amplitudes for the states after the potential is dropped.
u_{0}=(\frac{1}{\pi a^{2}})^{\frac{1}{4}} e^{{\frac{-x^2}{2a^2}}}
u_{1}=(\frac{4}{\pi})^{\frac{1}{4}} (\frac{1}{a^2})^\frac{3}{4} x e^{({\frac{-x^2}{2a^2}})}
u_{2}=(\frac{1}{4\pi a^2})^{\frac{1}{4}} (\frac{2x^2}{a^2}-1) e^{{\frac{-x^2}{2a^2}}}
Then, the potential is lowered suddenly, by setting \omega ' = \frac{\omega}{2}.
The system is then modeled with the expansion postulate, as a superposition of these three new states, with \omega' in them.
The particle was originally in the ground state before the potential was lowered, so I have that:
C_{i}=\int u_{i}^* \psi dx
Where \psi represents the ground state function, as stated above, with \omega. The states are real though, so the complex conjugate is unchanged from the normal state.
When I try to evaluate these integrals, I find that I can't actually do it. The problem is that the different symbols keep tripping me up... The ω and ω' values don't seem to gel.
Am I doing the physics wrong, or am I simple falling short on the maths?
Thanks, all.
Homework Statement
I have the first three states of the harmonic oscillator, and I need to know the amplitudes for the states after the potential is dropped.
Homework Equations
u_{0}=(\frac{1}{\pi a^{2}})^{\frac{1}{4}} e^{{\frac{-x^2}{2a^2}}}
u_{1}=(\frac{4}{\pi})^{\frac{1}{4}} (\frac{1}{a^2})^\frac{3}{4} x e^{({\frac{-x^2}{2a^2}})}
u_{2}=(\frac{1}{4\pi a^2})^{\frac{1}{4}} (\frac{2x^2}{a^2}-1) e^{{\frac{-x^2}{2a^2}}}
Then, the potential is lowered suddenly, by setting \omega ' = \frac{\omega}{2}.
The system is then modeled with the expansion postulate, as a superposition of these three new states, with \omega' in them.
The particle was originally in the ground state before the potential was lowered, so I have that:
C_{i}=\int u_{i}^* \psi dx
Where \psi represents the ground state function, as stated above, with \omega. The states are real though, so the complex conjugate is unchanged from the normal state.
The Attempt at a Solution
When I try to evaluate these integrals, I find that I can't actually do it. The problem is that the different symbols keep tripping me up... The ω and ω' values don't seem to gel.
Am I doing the physics wrong, or am I simple falling short on the maths?
Thanks, all.