- #1
Jeremiah Givens
- 8
- 1
Hello everyone,
I have been working through some research papers on a topic that really interests me, but I believe I am misunderstanding a few things about Dirac Notation. I have expressions that read:
\begin{align*}
&< \psi_n \mid g(H - E_{n+1}) \mid \psi_n> \text{,} \\
&< \psi_n \mid (H - E_{n})g(H - E_{n+1}) \mid \psi_n> \text{, and} \\
&< \psi_n \mid (H - E_{n})g(H - E_{n+1})g(H - E_{n}) \mid \psi_n>\text{,}
\end{align*}
where ##H## is the hamiltonian of the system, ##E_{i}## are just scalars, and ##g## is a function of position.
Now I thought that these expressions would be expanded out for calculations as follows:
\begin{align*}
< \psi_n \mid g(H - E_{n+1}) \mid \psi_n> &= \int \psi_n^* g(H - E_{n+1}) \psi_n d \tau \\
&= \int \psi_n^* g(H\psi_n - E_{n+1}\psi_n) d \tau \text{, and} \\
< \psi_n \mid (H - E_{n})g(H - E_{n+1}) \mid \psi_n> &= \int \psi_n^* (H - E_{n})g(H - E_{n+1}) \psi_n d \tau \\
&= \int \psi_n^* (H - E_{n})g(H\psi_n - E_{n+1}\psi_n) d \tau \\
&= \int \psi_n^* (H(g(H\psi_n - E_{n+1}\psi_n)) - E_{n}(g(H\psi_n - E_{n+1}\psi_n))) d \tau \text{, and finally}\\
< \psi_n \mid (H - E_{n})g(H - E_{n+1})g(H - E_{n}) \mid \psi_n> &= \int \psi_n^* (H - E_{n})g(H - E_{n+1})g(H - E_{n}) \psi_n d \tau \\
&= \int \psi_n^* (H - E_{n})g(H - E_{n+1})g(H\psi_n - E_{n}\psi_n) d \tau \\
&= \int \psi_n^* (H - E_{n})g(H(g(H\psi_n - E_{n}\psi_n)) - E_{n+1}(g(H\psi_n - E_{n}\psi_n))) d \tau \\
&= \int \psi_n^* (H(g(H(g(H\psi_n - E_{n}\psi_n)) - E_{n+1}(g(H\psi_n - E_{n}\psi_n)))) - E_{n}(g(H(g(H\psi_n - E_{n}\psi_n)) - E_{n+1}(g(H\psi_n - E_{n}\psi_n)))))d \tau \\
\end{align*}
So, essentially, my question is whether this is the proper way to expand these out? I have been trying to do calculations with these expressions, and I cannot get them to work, and I believe this is where my mistake is.
One of my professors said that he believes you're supposed to take the operator on the left and operate on the wave function on the left, and take the one one the right and operate on the wave function on the right, which seams a simpler, but I have been unable to find a text explaining why it is done this way. This also raises another question: what about when there are three operators? Do I act on the right with the right most operator, left with the left most, and then have a choice for the middle? Does the hamiltonian ever operate on the ##g## function?
If anybody could help me clear this up, it would be greatly appreciated!
Thanks,
Jere
I have been working through some research papers on a topic that really interests me, but I believe I am misunderstanding a few things about Dirac Notation. I have expressions that read:
\begin{align*}
&< \psi_n \mid g(H - E_{n+1}) \mid \psi_n> \text{,} \\
&< \psi_n \mid (H - E_{n})g(H - E_{n+1}) \mid \psi_n> \text{, and} \\
&< \psi_n \mid (H - E_{n})g(H - E_{n+1})g(H - E_{n}) \mid \psi_n>\text{,}
\end{align*}
where ##H## is the hamiltonian of the system, ##E_{i}## are just scalars, and ##g## is a function of position.
Now I thought that these expressions would be expanded out for calculations as follows:
\begin{align*}
< \psi_n \mid g(H - E_{n+1}) \mid \psi_n> &= \int \psi_n^* g(H - E_{n+1}) \psi_n d \tau \\
&= \int \psi_n^* g(H\psi_n - E_{n+1}\psi_n) d \tau \text{, and} \\
< \psi_n \mid (H - E_{n})g(H - E_{n+1}) \mid \psi_n> &= \int \psi_n^* (H - E_{n})g(H - E_{n+1}) \psi_n d \tau \\
&= \int \psi_n^* (H - E_{n})g(H\psi_n - E_{n+1}\psi_n) d \tau \\
&= \int \psi_n^* (H(g(H\psi_n - E_{n+1}\psi_n)) - E_{n}(g(H\psi_n - E_{n+1}\psi_n))) d \tau \text{, and finally}\\
< \psi_n \mid (H - E_{n})g(H - E_{n+1})g(H - E_{n}) \mid \psi_n> &= \int \psi_n^* (H - E_{n})g(H - E_{n+1})g(H - E_{n}) \psi_n d \tau \\
&= \int \psi_n^* (H - E_{n})g(H - E_{n+1})g(H\psi_n - E_{n}\psi_n) d \tau \\
&= \int \psi_n^* (H - E_{n})g(H(g(H\psi_n - E_{n}\psi_n)) - E_{n+1}(g(H\psi_n - E_{n}\psi_n))) d \tau \\
&= \int \psi_n^* (H(g(H(g(H\psi_n - E_{n}\psi_n)) - E_{n+1}(g(H\psi_n - E_{n}\psi_n)))) - E_{n}(g(H(g(H\psi_n - E_{n}\psi_n)) - E_{n+1}(g(H\psi_n - E_{n}\psi_n)))))d \tau \\
\end{align*}
So, essentially, my question is whether this is the proper way to expand these out? I have been trying to do calculations with these expressions, and I cannot get them to work, and I believe this is where my mistake is.
One of my professors said that he believes you're supposed to take the operator on the left and operate on the wave function on the left, and take the one one the right and operate on the wave function on the right, which seams a simpler, but I have been unable to find a text explaining why it is done this way. This also raises another question: what about when there are three operators? Do I act on the right with the right most operator, left with the left most, and then have a choice for the middle? Does the hamiltonian ever operate on the ##g## function?
If anybody could help me clear this up, it would be greatly appreciated!
Thanks,
Jere
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