SHM Mix of States (3-325)

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


valentin68

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