- #1
xicor
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Homework Statement
Find the following hermitian conjugates and show if they are hermitian operators:
i) xp
ii) [x , p]
iii) xp + px
Where x is the position operator and p is the momentum operator.
Homework Equations
<f|Qg> = <Q[itex]^{t}[/itex]f|g>
Q = Q[itex]^{t}[/itex] Hermitian operator
p = -ih(d/dx)
The Attempt at a Solution
For the first case I have applied the condition of hermitian operators where I get <f|-ihx(dg(x)/dx)> and then get the form <f|-ih(dxg(x)/dx)>which leads to the integral ∫f*(-ih(d/dx)(xg(x)dx but am not sure how you bring the operator to f*. Do you just do the product rule and bring the product or to f* or do you need to do integration by part?
For the case of [x , p] you get [x, p] = -xih(d/dx) +ih(dx/dx) and find I apply this to the hermitian conjugate I get <-ihf|g> but was told something was wrong since I got it from <-xih(dg*/dx)+ih(g +x(dg/dx)>. For the third part I'm not getting anything that make sense when I apply -(xih(d/dx) + ih(dx/dx)) and got ∫f*(ih(g + 2x(dg/dx))dx but according to my notes xp + px should be hermitian.
Thanks to anybody that helps.