I Inner products with spherical harmonics in quantum mechanics

Salmone
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Let ##|l,m\rangle## be a simultaneous eigenstate of operators ##L^2## and ##L_z## and we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##. It is true that in general ##\langle l,m|cos(\theta)|l,m'\rangle=0## ##(1)## for the same ##l## even if m's are equal? Why?

For example ##\langle1,0|cos(\theta)|1,0\rangle= \int_{}^{} (Y_1^0)^{*}cos(\theta)Y_1^0 \,d\Omega ## is this zero? Does the fact that equation (1) applies have anything to do with parity?
 
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Salmone said:
we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##.
What does this even mean? ##\cos(\theta)## isn't an operator.
 
Of course ##\cos \theta## is an operator. The trick to get the matrix elements wrt. the angular-momentum eigenstates is to use the socalled Wigner Eckart theorem to evaluate matrix elements of tensor-operator components. In your case you have ##z=r \cos \theta##, i.e., you can evaluate the corresponding expectation values considering the position-vector components ##\vec{x}##, using the Wigner-Eckart theorem.

https://en.wikipedia.org/wiki/Wigner–Eckart_theorem
 
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It isn't an operator in the form in which the OP used it, i.e. on abstract vectors in Hilbert space.

To calculate the integral in the OP I would write ##\cos\theta## in terms of spherical harmonics. Then you can look up an expansion for the product of two spherical harmonics in terms of a sum of single spherical harmonics (involves Wigner 3j -symbols which you'll need to understand the properties of). From there you can compute the integral using the orthogonality of spherical harmonics.
 
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That's of course another possibility and equivalent with the use of the Wigner-Eckart theorem :-).
 
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Thank you for all your answers, I'd like to know how to say instantly that ##\langle lm|cos(\theta)|lm'\rangle=0## using the parity where ##|lm\rangle=Y_l^m(\theta,\phi)##. Is there a way to prove it? By "using parity" I mean something like: spherical harmonics parity is equal to ##(-1)^lY_l^m## and ##cos(\pi-\theta)=-cos(\theta)##
 
Since ##\cos \theta=z/r## the parity of this operator is ##(-1)##, because the position vector is a polar vector and ##r=|\vec{x}|## is a scalar. Thus the parity of ##\cos \theta Y_{l}^m## is ##(-1)^{l+1}##. Since the spherical harmonics are parity eigenstates, the matrix element is the same ##l## must vanish (or with different ##l## of the same parity, i.e., with both ##l## even or both odd).
 
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@vanhees71 I can't get the last passage, why the matrix element in the same ##l## must vanish? If possible, could you be more explicit please?
 
Obviously ##\cos \theta Y_{l}^m(\theta,\phi)## has the opposite parity than ##Y_{l}^{m'}(\theta,\phi)##. That implies that the integrand in the integral of the matrix element is antisymmetric under ##\theta \rightarrow \pi-\theta##, which makes the integral over ##\theta \in [0,\pi]## vanish.

Indeed if ##f(\theta)=-f(\pi-\theta)## you have on the one hand
$$I=\int_0^{\pi} \mathrm{d} \theta \sin \theta f(\theta).$$
Now substitute ##\theta'=\pi-\theta##
$$I=\int_0^{\pi} \mathrm{d} \theta' \sin(\pi-\theta') f(\pi-\theta').$$
Now ##\sin(\pi-\theta')=\sin \theta'## and by assumption ##f(\pi-\theta')=-f(\theta')##, i.e.,
$$I=-\int_0^{\pi} \mathrm{d} \theta' \sin \theta' f(\theta')=-I \; \Rightarrow \; I=0.$$
 
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Salmone said:
For example ##\langle L,0|cos(\theta)L,0\rangle= \int_{}^{} (Y_L^0)^{*}cos(\theta)Y_L^0 \,d\Omega ## is this zero?
It's just simple integration. After the substitution ##z=\cos\theta##, the integral becomes ##\int_{-1}^{+1}dz## of an odd function of z which vanishes.
 
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  • #11
The scalar product for functions on the unit sphere is defined by
$$\langle f|g \rangle=\int_0^{\pi} \mathrm{d} \theta \int_0^{2 \pi} \mathrm{d} \phi \sin \theta f^*(\theta,\phi) g(\theta,\phi).$$
In my calculation
$$f(\theta)=Y_l^{m*}(\theta,\varphi) Y_l^{m'}(\theta,\varphi) \cos \theta.$$
The integral over ##\theta## alone makes the integral vanish, no matter what the ##\phi## dependence is. The ##\sin \theta## must be in the integral because of the definition of the scalar product for ##\mathrm{L}^2(\Omega)##, where ##\Omega## is the unit sphere in ##\mathbb{R}^3##.
 
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One last thing, is it possible to write the integral as ##\int_{-1}^{1} \mathrm{d} cos\theta \int_0^{2 \pi} \mathrm{d} \phi f^*(\theta,\phi) g(\theta,\phi)## so that I'm integrating from ##-1## to ##1## an odd function?
 
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  • #14
Sure. Just substitute ##u=\cos \theta##. Then ##\mathrm{d} u =-\mathrm{d} \theta \sin \theta## and thus
$$\langle f|g \rangle = \int_{-1}^1 \mathrm{d} u \int_0^{2 \pi} \mathrm{d} \phi f^* g,$$
where I wrote in a somewhat sloppy notation.
 
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