Question

3) A harmonic oscillator is in the state of wavefunction \begin{equation} \Psi(x, 0) = 2Ai\phi_0 - \frac{1}{2}e^{2i\alpha}\phi_1 \end{equation} where $\phi_0$ and $\phi_1$ are orthonormal functions. a) Find A b) Find $\Psi(x, t)$ c) Calculate the expectation value of $< a_+ >$ and, $< a_- >$ in $\Psi(x, t)$ d) Calculate the expectation value of position $< x >$ and momentum $< p >$ in $\Psi(x, t)$ e) Calculate the expectation value of $< x^2 >$ and $< p^2 >$ in $\Psi(x, t)$.

          3) A harmonic oscillator is in the state of wavefunction
\begin{equation}
\Psi(x, 0) = 2Ai\phi_0 - \frac{1}{2}e^{2i\alpha}\phi_1
\end{equation}
where $\phi_0$ and $\phi_1$ are orthonormal functions.
a) Find A
b) Find $\Psi(x, t)$
c) Calculate the expectation value of $< a_+ >$ and, $< a_- >$ in $\Psi(x, t)$
d) Calculate the expectation value of position $< x >$ and momentum $< p >$ in $\Psi(x, t)$
e) Calculate the expectation value of $< x^2 >$ and $< p^2 >$ in $\Psi(x, t)$.

3) A harmonic oscillator is in the state of wavefunction

Ψ(x, 0) = 2Aiϕ0 - (1)/(2)e^2iαϕ1

where ϕ0 and ϕ1 are orthonormal functions.
a) Find A
b) Find Ψ(x, t)
c) Calculate the expectation value of < a+ > and, < a- > in Ψ(x, t)
d) Calculate the expectation value of position < x > and momentum < p > in Ψ(x, t)
e) Calculate the expectation value of < x^2 > and < p^2 > in Ψ(x, t).

Added by Erica S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Shaiju T

06/03/2022

Step 1

The normalization condition is given by: ∫ |Y(x,0)|^2 dx = 1 Since Y(x,0) = 2Ai$o, we have: ∫ |2Ai$o|^2 dx = 1 Simplifying, we get: ∫ 4A^2|i$o|^2 dx = 1 Since |i$o|^2 = 1 (since i$o is an orthonormal function), we have: ∫ 4A^2 dx = 1 Integrating, we Show more…

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3) A harmonic oscillator is in the state of wavefunction Y(x,0)=2Ai$o where o and , are orthonormal functions. a) Find A b) Find Y(x,t)) c) Calculate the expectation value of < a+ > and ,< a_ > in Y(x, t)) d) Calculate the expectation value of position < x > and momentum < p > in Y(x,t)) e) Calculate the expectation value of < x2 > and < p2 > in Y(x, t)).