- #1
Keru
- 20
- 1
"Given a 3D Harmonic Oscillator under the effects of a field W, determine the matrix for W in the base given by the first excited level"
So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result:
W = 2az2 - ax2 - ay2 + 2az+ 2 -ax+ 2 -ay+ 2 + 2{az, az+}) - {ax, ax+}) - {ay, ay+}
As for the following part, I've proceeded like this, which I'm pretty sure it's wrong at some point.
In terms of the ladder operator, I'm using the basis:
|nx, ny,nz>
With ni corresponding to the level of excitement on a certain coordinate. I have taken the first excited level to be:
|1,0,0>
I generally find the matrix elements like:
Mi,j = <i|M^|j>
So my thought has been, I have to find them in the basis |1,0,0>, let's just:
<1,0,0|W|1,0,0>
Which looks terrible because that's clearly a 1x1 "matrix".
What am I missing? Do the matrix actually correspond to |1,0,0>, |0,1,0>, |0,0,1> since all these possible combinations correspond to the first energy state? Is that simply not the basis I should be using?
Any help would be greatly appreciated.
So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result:
W = 2az2 - ax2 - ay2 + 2az+ 2 -ax+ 2 -ay+ 2 + 2{az, az+}) - {ax, ax+}) - {ay, ay+}
As for the following part, I've proceeded like this, which I'm pretty sure it's wrong at some point.
In terms of the ladder operator, I'm using the basis:
|nx, ny,nz>
With ni corresponding to the level of excitement on a certain coordinate. I have taken the first excited level to be:
|1,0,0>
I generally find the matrix elements like:
Mi,j = <i|M^|j>
So my thought has been, I have to find them in the basis |1,0,0>, let's just:
<1,0,0|W|1,0,0>
Which looks terrible because that's clearly a 1x1 "matrix".
What am I missing? Do the matrix actually correspond to |1,0,0>, |0,1,0>, |0,0,1> since all these possible combinations correspond to the first energy state? Is that simply not the basis I should be using?
Any help would be greatly appreciated.