Problem 3¶
Angular Eigenstate of the SHO
In your lecture notes you have seen the mathematical derivation of an angular momentum eigenstate of the 2D simple harmonic oscillator, in the case where the potential is spherically symmetric. This eigenstate a superposition of the first two excited energy eigenstates such that
Here unx,ny(x,y)=unx(x)×uny where unx is an energy eigenfunction of the 1D SHO.
Assume an electron is confined in a simple harmonic potential well with ω0=2×1016 s−1. Produce an animation showing the time evolution of the energy eigenstates u10(x,y,t) and iu01(x,y,t). What effect does the factor of i have on the eigenstate iu01?
Plot an animation showing the real part of the full time dependent wavefunction.
The following code can be used to create an animated contour plot:
#First set up the figure
fig, ax = plt.subplots(1, 1)
ax1.set_xlim((x[0],x[-1]))
ax1.set_ylim((y[0], y[-1]))
quad1 = ax1.pcolormesh(x, y, z, shading='gouraud',
vmin=<min z>, vmax=<max z>) # this line is vital to ensure the colour map stays consistent
def init():
quad1.set_array([])
return quad1
def animate(i):
z = <z at time t>
quad1.set_array(z.ravel())
return quad1
anim = animation.FuncAnimation(fig, animate)
plt.close(fig) # Include this line to prevent two figures appearing in the notebook
HTML(anim.to_jshtml()) # This line allows you to see the animation in the notebook