Exploring the Applications of Dirac Form in Quantum Mechanics

[johanjones190]

I am studying for a Quantum Mechanics final and our prof. gave us an equations sheet with some of the equations needed for the exam.
I was wondering what the following equations could be used for. We have covered spherical harmonics, the Hydrogen Atom, Degenerate Perturbation Theory, Spin, Stationary Electron in a Magnetic field, Addition of Angular Momentum (J, Jz) using Clebsch Gordon table, and Spin Orbit and the Zeeman Effect. I am a little confused by the dirac form.

Sorry about the pdf form... it wouldn't let me paste the code into the text!

Thanks

 

[lbrits]

What he has written down is in fact $$\left\langle \mathbf{r} | 0 1 \right\rangle$$
Presumably, you could use these equations to find, say, $$\left\langle y^2 \right\rangle$$ for the 2D harmonic oscillator.

 

[johanjones190]

I am not familiar with that form... is that just Psi(r)? and what quantum numbers are the 0 and 1?
Thanks again!

 

[lbrits]

Mmm... it would be a good idea for you to review Bra/ket notation. The state in this case is one of the excited states of the quantum harmonic oscillator:

$$\left| n_x = 0, n_y = 1\right\rangle$$.

I'm talking about the overlap of that state with $$\left|\mathbf{r}\right\rangle$$, which is an eigenstate of the position operator with eigenvalue $$\mathbf{r}$$. Thus I'm referring to the wavefunction in the position representation.

 

[johanjones190]

Thanks, I haven't done much with Bra/ket notation!