These videos show how raising and lowering operators are defined and used for the one-dimensional harmonic oscillator (1DHO). It uses the same notation as Griffiths, chapter 2.
The two key equations you need in order to use the operators are:
\(a_- | n \rangle = \sqrt{n}\: |n-1 \rangle \),
\(a_+ | n \rangle = \sqrt{n+1} \: |n+1 \rangle \).
This is shown at about 24:25 of the third video (using a- a+). Everything before that is derivation.
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Midterm 2 solutions
Here are solutions to midterm 2. I see that I did not write it in the solutions, but it is helpful for sp2 type integrals to recall that \(\...
I think the residues are sqrt(n) for a- and sqrt(n+1) for a+. Easy thing to forget, I did the same thing on my first try.
ReplyDeleteAgree! Is there a specific mistake in a video on that?
DeleteI see what you mean now. In the last video there is a mistake.
DeleteHopefully everyone can find that. In the evaluation of the 2|2 term.
DeleteOh, so we multiply by the square root of n or (n+1), not just multiply by n or (n+1) like its done in the third video?
DeleteThe two key equations you need in order to use the operators are:
Delete\(a_- | n \rangle = \sqrt{n}\: |n-1 \rangle \),
\(a_+ | n \rangle = \sqrt{n+1} \: |n+1 \rangle \).
This is shown at about 24:25 of the third video (using a- a+). Everything before that is derivation.
Hi Nick. I am not sure exactly where you mean in the third video. The part at 24:25 is correct and uses The two key equations you need in order to use the operators are:
Delete\(a_- | n \rangle = \sqrt{n}\: |n-1 \rangle \),
\(a_+ | n \rangle = \sqrt{n+1} \: |n+1 \rangle \).
so, yes, you do want the square root.
DeleteIn the second video the part where the lowering operator is operating on /psi_{n} is it supposed to be equal to c_{n}* | /psi_{n+1} > or c_{n+1}* | /psi_{n+1} >
ReplyDeleteI ask because in the previous line it is c_{n+1}* | /psi_{n+1} >
DeleteThe two key equations you need in order to use the operators are:
Delete\(a_- | n \rangle = \sqrt{n}\: |n-1 \rangle \),
\(a_+ | n \rangle = \sqrt{n+1} \: |n+1 \rangle \).
This is shown at about 24:25 of the third video (using a- a+). Everything before that is derivation.