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Physics
Quantum Physics
Finding matrices of perturbation using creation/annihilation operators
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[QUOTE="Keru, post: 6331773, member: 482767"] "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 = 2a[SUB]z[/SUB][SUP]2[/SUP] - a[SUB]x[/SUB][SUP]2[/SUP] - a[SUB]y[/SUB][SUP]2[/SUP] + 2a[SUB]z[/SUB][SUP]+ 2[/SUP] -a[SUB]x[/SUB][SUP]+ 2[/SUP] -a[SUB]y[/SUB][SUP]+ 2[/SUP] + 2{a[SUB]z[/SUB], a[SUB]z[/SUB][SUP]+[/SUP]}) - {a[SUB]x[/SUB], a[SUB]x[/SUB][SUP]+[/SUP]}) - {a[SUB]y[/SUB], a[SUB]y[/SUB][SUP]+[/SUP]} 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: |n[SUB]x[/SUB], n[SUB]y[/SUB],n[SUB]z[/SUB]> With n[SUB]i[/SUB] 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: M[SUB]i,j[/SUB] = <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. [/QUOTE]
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Finding matrices of perturbation using creation/annihilation operators
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