Compute the uncertainty product for a harmonic oscillator
Not what you're looking for?
See attached file.
Purchase this Solution
Solution Summary
A detailed solution is derived from first principles.
Solution Preview
Notation: If A is an operator then I'll denote the Hermitian conjugate of A by C(A)
The annihilation operator in terms of p and x is:
a = 1/sqrt(2m hbar omega) [m omega x + i p]
And the creation operator is its Hermitian conjugate:
C(a) = 1/sqrt(2m hbar omega) [m omega x - i p]
It is convenient to have the expressions for x and p in terms of a and C(a). Solving the above equations for x and p gives:
x = Squareroot[h-bar/(2 m omega)] [a + C(a)]
p = i Squareroot[m omega h-bar/2] [C(a) - a]
The Hamiltonian is:
H = [C(a) a + 1/2] hbar omega
a and C(a) satisfy the commutation relation:
[a, C(a)] = 1 (1)
We also have:
[H, a] = [[C(a) a + 1/2] hbar omega, a] = hbar omega [C(a) a, a] = hbar omega [C(a)a a - a C(a) a ]
From (1) it follows that a C(a) = 1 + C(a) a. The second term in the above formula can thus be written as:
a C(a) a = [1 + C(a) a] a = a + C(a) a a
The expression for the commutator thus simplifies to:
[H, a] = -hbar omega a (2)
We can compute the eigenvectors and eigenvalues of H as follows. We put C(a) a = N. Then since H = hbar omega(N+1/2)
we can just as well compute the eigenstates of N. These are then also the eigenstates of H. If |n> is a normalized eigenstate of N with eigenvalue n, then N|n> = n|n> and thus H|n> = (n+ 1/2)hbar omega|n>, so the corresponding eigenvalue of H is
(n + 1/2)hbar omega. Note that we are not assuming that n is an integer.
Consider the action of N on the state a|n>:
N a|n> = C(a) a a|n>
Eq. (1) says that
a C(a) - C(a) a = 1 ------>
C(a) a = a C(a) - 1
and we see that
N a|n> = C(a) a a|n> = [a C(a) - 1]a |n> = a C(a) a|n> - a|n> = a N|n> - a|n> = n a |n> - a |n> = (n-1)|n> (3)
So, it looks like a|n> is an eigenstate with eigenvalue n-1. Let's compute the norm of this state. The absolute value squared of the norm is:
<n|C(a) a|n> = <n|N|n> = <n|n|n> = n <n|n> = n. (4)
The absolute value squared of any ...
Purchase this Solution
Free BrainMass Quizzes
Introduction to Nanotechnology/Nanomaterials
This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.
Basic Physics
This quiz will test your knowledge about basic Physics.
Intro to the Physics Waves
Some short-answer questions involving the basic vocabulary of string, sound, and water waves.
Classical Mechanics
This quiz is designed to test and improve your knowledge on Classical Mechanics.
The Moon
Test your knowledge of moon phases and movement.