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Context: (not asking for solution)

I'm attempting to show $\langle n,l',m'|\hat z|n,l,m \rangle = 0$ for $m\neq m'$ using the explicit form of $Y_{l,m}(\theta,\phi)$.

Question:

I wasn't sure how to handle the position operator $\hat z$, so my idea was to express it in spherical coordinates.

It is not clear to me if I'm allowed to do something like $\hat z = \hat r \cos(\theta)$

Do position operators work like this?

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    $\begingroup$ This would suggest that the radial unit vector $\hat{r}$ is always in the $z$-direction, which can't be true. See here for how to go between Cartesian and spherical unit vectors. $\endgroup$
    – march
    Commented Feb 20, 2020 at 1:48
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    $\begingroup$ You can also try to express the spherical harmonics in Cartesian. $\endgroup$ Commented Feb 20, 2020 at 3:46
  • $\begingroup$ @march I may be confused but I was under the impression that the position operator $/hat x$ was different than the unit vector $/hat x$ $\endgroup$ Commented Feb 20, 2020 at 4:24
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    $\begingroup$ I see. I misunderstood the hat to be indicating a unit vector rather than an operator. (That always gets me). Yes, then you are correct. $\endgroup$
    – march
    Commented Feb 20, 2020 at 4:57

1 Answer 1

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You can write $z$ as

$$ z = r\left(\frac{4\pi}{3}\right)^{1/2} Y_{1,0}(\theta,\phi) $$

This amounts to evaluating the matrix element in the following way

$$ \langle n\;l^\prime m^\prime | z | nlm\rangle = \sqrt{\frac{4\pi}{3}}\int R_{n^\prime l^\prime}(r)R_{nl}(r)r^3\,dr\int Y_{l^\prime m^\prime}(\theta, \phi)Y_{10}(\theta, \phi)Y_{lm}(\theta, \phi)\,d\Omega$$

I carried out the full calculation just to give an idea of how to evaluate this integral in the case of linear polarization on the $z$ axis.

Since we are only interested into the projection of the angular momentum, we just have to consider the part of the spherical harmonics which depends upon $m$. This part, in the integral, is given by the product of the three exponential factors of the three spherical harmonics which depend on $\phi$, so $$ Y_{l^\prime m^\prime}Y_{1q}Y_{lm} \propto e^{-im^\prime \phi}e^{iq\phi}e^{im\phi}$$ in our case $q=0$. The integral in $\phi$ is non-vanishing only when $m^\prime = m+q$ which in our case amounts as saying that the integral is non-vanishing only is $m^\prime = m$

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  • $\begingroup$ There's too much detail here. We don't provide complete answers to homework-like questions. They don't do the OP much good. $\endgroup$
    – garyp
    Commented Feb 20, 2020 at 11:31
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    $\begingroup$ @garyp Yes, i understand, but this calculations can easily be found everywhere, even in many books. I thought that giving an example to the question asked by the OP would help him understand the basics so that he could do the other calculations in a similar manner. I'll edit my answer anyway $\endgroup$ Commented Feb 20, 2020 at 11:35

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