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flyusx
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- Homework Statement
- Are ##\hat{A}=\frac{\partial}{\partial\phi}## and ##\hat{B}=\frac{\partial}{\partial\theta}## Hermitian? Given ##\psi(\theta,\phi)=\exp(-3i\phi)\cos(\theta)##, evaluate <ψ|Α|ψ>, <ψ|Β|ψ>
- Relevant Equations
- Hermitian functions can act either on the bra or ket first; non-Hermitian functions do not have tis ability. <φ|ψ>##=\int\phi^{*}\psi##
I know the way to solve the first part is to find <ψ|Αψ> and compare it with <ψΑ|ψ>. This comparison can be done through an integral representation where we take ψ* and act A on ψ to be the integrand, or act A on ψ* and multiply by ψ for the integrand. If the integrals are the same, then the operator is Hermitian. (Another way is to check the eigenvalues and see if they are purely real). My question is: after computing the integrand, do we expand the integral out as a double integral with respect to dφdθ? Or do we only expand it in terms of dθ since the exponential term cancels out? I assume the bounds for the integral will be from 0 to 2π as this would represent a full 'declination and ascension' over space. Knowing this will help me solve both parts.
If it's of any use, this is Exercise 2.52 of Zettili's Quantum Mechanics. I referred to his solved Problem 2.19 to see the best way to represent an inner product, but the wave functions there were solely dependent on x and thus integrated with respect to dx. I couldn't find an example where an inner product of a multivariable wave function was expanded out in integral form.
If it's of any use, this is Exercise 2.52 of Zettili's Quantum Mechanics. I referred to his solved Problem 2.19 to see the best way to represent an inner product, but the wave functions there were solely dependent on x and thus integrated with respect to dx. I couldn't find an example where an inner product of a multivariable wave function was expanded out in integral form.
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