Consider a system described by a Hamiltonian $H=p/^22+x^2/2$ which is written in the system of units such that $\hbar=1$, and $p=-i*(d/dx)$. At $t=0$ the system is in the state described by the wave function $\psi(x,0)=1/\sqrt{8\pi}\phi_0(x)+1/\sqrt{18\pi}\phi_2(x)$, where $\phi_0=e^{-x^2/2}$ and $\phi_2(x)=(1-2x^2)*e^{-x^2/2}$ are non-normalized eigenfunctions of the Hamiltonian.

a) Find the eigenvalues of the Hamiltonian corresponding to $\phi_0(x)$, $\phi_2(x)$

b) Normalize the eigenfunctions and rewrite the initial state in terms of the normalized eigenfunctions.

c) Find the expression for the wavefunction at a latter time $t>0$

d) If the energy is measured, which values can be observed, and what are their probabilities?

$H ̂=p ̂^2/2+x ̂^2/2$

a)

For the given wavefunctions

$ϕ_0=e^{-x^2/2}$ and $ϕ_2=(1-2x^2 )*e^{-x^2/2}$

$E_0=<ϕ_0 |H| ϕ_0>=$

$=-1/2*∫_{-∞}^∞ e^{-x^2/2}*d^2/(dx^2 ) (e^{-x^2/2} )*dx+1/2*∫_{-∞}^∞ e^{-x^2/2}*x^2*e^{-x^2/2}*dx$

or

$E_0=-1/2*∫_{-∞}^∞ (x^2-1)*e^{-x^2}*dx+1/2*∫_{-∞}^∞ x^2 e^{-x^2} dx=$

$=(-1/2)(-√π/2)+1/2*√π/2=√π/2$

$E_2=<ϕ_2 |H| ϕ_2>=$

$=-1/2 ∫_{-∞}^∞ (1-2x^2 ) e^{-x^2/2}*d^2/(dx^2 ) [(1-2x^2 ) e^{-x^2/2}]dx+1/2 ∫_{-∞}^∞ x^2 (1-2x^2 )^2 *e^{-x^2} dx$

or

$E_2=-1/2 ∫_{-∞}^∞ (1-2x^2)[8x^2+(1-2x^2 )(x^2-1)-4] *e^{-x^2}*dx+1/2*5√π$

$E_2=-1/2*(-5√π)+1/2*5√π=5√π$

b)

$∫_{-∞}^∞ |ϕ_0 |^2 dx=∫_{-∞}^∞ e^{-x^2} dx=√π$

so that $ϕ_{0n}=1/(π)^(1/4) *ϕ_0$

$∫_{-∞}^∞ |ϕ_2 |^2 dx=∫_{-∞}^∞ (1-2x^2 )^2*e^{-x^2} dx=2√π$

so that $ϕ_{2n}=1/(4π)^{1/4} *ϕ_2$

Therefore

$ψ(x,0)=(1/√8π) ϕ_0+(1/√18π) ϕ_2=(π)^{1/4}/(8π)^{1/2} ϕ_0n+(4π)^{1/4}/(18π)^{1/2} *ϕ_{2n}$

c)

At $t>0$ (because the potential is time independent)

$ψ(x,t)=(1/√8π) ϕ_0 (x)*e^{-(i*E_0/ℏ)*t}+(1/√18π) ϕ_2 (x)*e^{-(iE_2/ℏ)*t}$ with $ℏ=1$

$ψ(x,t)=(1/√8π) ϕ_0 (x)*e^{-(i√π*t)/2}+(1/√18π) ϕ_2 (x)*e^{-i*5t√π}$

d)

Since $ϕ_0$ and $ϕ_2$ are eigenfunctions they are orthogonal to each other. This means

$<ψ|ψ>=1/8π+1/18π=13/72π$

The values of the energy that can be observed are the two eigenvalues found above

$E_0=√π/2$ and $E_2=5√π$

With the probabilities

$P_0=(|<ϕ_0 |ψ>|^2)/(<ψ|ψ>)=(1/8π)/(13/72π)=9/13$

and $P_2=(|<ϕ_2 |ψ>|^2)/(<ψ|ψ>)=(1/18π)/(13/72π)=4/13$