How Do Quantum Harmonic Oscillator Ladder Operators Affect State Vectors?

[glederfein]

Homework Statement

Given a quantum harmonic oscillator, calculate the following values:
[itex]\left \langle n \right | a \left | n \right \rangle, \left \langle n \right | a^\dagger \left | n \right \rangle, \left \langle n \right | X \left | n \right \rangle, \left \langle n \right | P \left | n \right \rangle[/itex]

Homework Equations

Hamiltonian: [itex]H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2X^2[/itex]
Ladder operators:
[itex]a=\sqrt{\frac{m\omega}{2\hbar}}\left ( X + \frac{i}{m\omega}P \right )[/itex]
[itex]a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\left(X-\frac{i}{m\omega}P \right )[/itex]
[itex]\left [ a,a^\dagger \right ] = 1[/itex]
[itex]\left [ a^\dagger a,a \right ] = -a[/itex]
N operator:
[itex]N=a^\dagger a[/itex]
[itex]N\left | n \right \rangle = n\left | n \right \rangle[/itex]
[itex]a^\dagger \left | n \right \rangle = \sqrt{n+1} \left | n+1 \right \rangle[/itex]
[itex]a \left | n \right \rangle = \sqrt{n} \left | n-1 \right \rangle[/itex]
[itex]\left | n \right \rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} \left | 0 \right \rangle[/itex]

The Attempt at a Solution

[itex]\left \langle n \right | a \left | n \right \rangle = \left \langle n \right | \sqrt{n} \left | n-1 \right \rangle =
\sqrt{n} \left \langle n | n-1 \right \rangle =
\sqrt{n} \left ( \left | n \right \rangle \right ) ^ \dagger \left | n-1 \right \rangle =
\sqrt{n} \left ( \frac{(a^\dagger)^n}{\sqrt{n!}} \left | 0 \right \rangle \right ) ^ \dagger \frac{(a^\dagger)^{n-1}}{\sqrt{(n-1)!}} \left | 0 \right \rangle =
\sqrt{\frac{n}{n!(n-1)!}} \left \langle 0 \right | a^n (a^\dagger)^{n-1} \left | 0 \right \rangle[/itex]

Not sure how to continue from here...

 

[TSny]

The Attempt at a Solution

[itex]\left \langle n \right | a \left | n \right \rangle = \left \langle n \right | \sqrt{n} \left | n-1 \right \rangle =
\sqrt{n} \left \langle n | n-1 \right \rangle =
\sqrt{n} \left ( \left | n \right \rangle \right ) ^ \dagger \left | n-1 \right \rangle =
\sqrt{n} \left ( \frac{(a^\dagger)^n}{\sqrt{n!}} \left | 0 \right \rangle \right ) ^ \dagger \frac{(a^\dagger)^{n-1}}{\sqrt{(n-1)!}} \left | 0 \right \rangle =
\sqrt{\frac{n}{n!(n-1)!}} \left \langle 0 \right | a^n (a^\dagger)^{n-1} \left | 0 \right \rangle[/itex]
.

When you get to [itex]\left \langle n \right | a \left | n \right \rangle = \left \langle n \right | \sqrt{n} \left | n-1 \right \rangle =
\sqrt{n} \left \langle n | n-1 \right \rangle [/itex] you should be able to see the answer.

 

[glederfein]

When you get to [itex]\left \langle n \right | a \left | n \right \rangle = \left \langle n \right | \sqrt{n} \left | n-1 \right \rangle =
\sqrt{n} \left \langle n | n-1 \right \rangle [/itex] you should be able to see the answer.

Is the answer zero because eigenvectors are always perpendicular to one another?
Doesn't that mean that all the four values in the question are zero?

 

[TSny]

Yes, eigenvectors of a Hermitian operator that correspond to different eigenvalues are orthogonal.

And, yes, all 4 are zero

 

[paranormal]

Firstly, it is important to note that the quantum harmonic oscillator is a fundamental model in quantum mechanics and is used to describe a wide range of physical systems, such as atoms, molecules, and solid state materials.

To calculate the values given in the homework statement, we can use the ladder operators and the N operator as shown in the equations above. Using these operators, we can rewrite the given expressions as follows:

\left \langle n \right | a \left | n \right \rangle = \sqrt{n} \left \langle n | n-1 \right \rangle = \sqrt{n} \left ( \sqrt{n-1} \left \langle n | n-2 \right \rangle \right ) = \sqrt{n(n-1)} \left \langle n | n-2 \right \rangle = ... = \sqrt{\frac{n!}{(n-n)!}} \left \langle n | 0 \right \rangle

Similarly, we can rewrite the other expressions as:

\left \langle n \right | a^\dagger \left | n \right \rangle = \sqrt{n+1} \left \langle n | n+1 \right \rangle = \sqrt{(n+1)n} \left \langle n | n+2 \right \rangle = ... = \sqrt{\frac{(n+1)!}{(n+n+1)!}} \left \langle n | n+n+1 \right \rangle

\left \langle n \right | X \left | n \right \rangle = \left \langle n \right | \sqrt{\frac{\hbar}{2m\omega}} (a+a^\dagger) \left | n \right \rangle = \sqrt{\frac{\hbar}{2m\omega}} \left ( \left \langle n | a \left | n \right \rangle + \left \langle n \right | a^\dagger \left | n \right \rangle \right ) = \sqrt{\frac{\hbar}{2m\omega}} \left ( \sqrt{n} \left \langle n | n-1 \right \rangle + \sqrt{n+1} \left \langle n | n+1 \right \rangle \right )

\left \langle n \right | P \left | n \right \rangle = \left \langle n \right | \sqrt