Lab 8 - Simple Harmonic Oscillator states - Quantum Mechanics with QuTiP

Problems from Chapter 12

from numpy import sqrt
from qutip import *

Define the standard operators¶

N = 10  # pick a size for our state-space
a = destroy(N)
n = a.dag()*a

Problem 12.1:¶

a*a.dag() - a.dag()*a

Problem 12.2:¶

n*a.dag() - a.dag()*n

n*a.dag() - a.dag()*n == a.dag()

Problem 12.3 (use n=2 as a test-case):¶

To define $|2\rangle$ use the basis(N,n) command where N is the dimension of the vector, and n is the quantum number.

psi = basis(N,2)

psi

a.dag()*psi

a.dag()*basis(N,2) == sqrt(3)*basis(N,3)

Problem 12.5 and 12.6:¶

These are simple, just view the matrix representation of the operators

a.dag()

Problem 12.7:¶

First, define $\hat{X}$ and $\hat{P}$ operators

X = 1/2 * (a + a.dag())
P = 1/2j * (a - a.dag())

psi = 1/sqrt(2)*(basis(N,1)+basis(N,2))
ex = psi.dag()*X*psi  # or expect(X, psi)
exq = psi.dag()*X*X*psi  # expect(X*X, psi)
ep = psi.dag()*P*psi  # expect(P, psi)
epq = psi.dag()*P*P*psi  # or expect(P*P, psi)

deltaX = sqrt(exq - ex**2)
deltaP = sqrt(epq - ep**2)

deltaX * deltaP * 2 # compare to uncertainty relation (ΔxΔp >= 1/2)
# the factor of two is to convert from the unitless version of the operator

Alternatively, we can find the indeterminacy bound for ΔX and ΔP (the unitless operators):

\Delta X \Delta P \geq \frac{1}{2}\left|\left\langle \left[\hat{X},\hat{P}\right] \right\rangle\right|

(1)

1/2*(psi.dag()*commutator(X,P)*psi)

Which is also satisfied by the calculated value (1.41 > 0.25)

Problem 12.8:¶

psi = 1/sqrt(2)*(basis(N,2)+basis(N,4))
ex = expect(X, psi)
exq = expect(X*X, psi)
ep = expect(P, psi)
epq = expect(P*P, psi)

deltaX = sqrt(exq - ex**2)
deltaP = sqrt(epq - ep**2)

deltaX * deltaP * 2 # to compare to book solution which uses the full x and p operators with units