LightPhoton
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In A Modern Approach to Quantum Mechanics, Townsend writes:
One of the most evident features of the position-space representations (9.117), (9.127), and (9.128) of the angular momentum operators is that they depend only on the angles ##\theta## and ##\phi##, not at all on the magnitude ##r## of the position vector. Rotating a position eigenstate changes its direction but not its length. Thus we can isolate the angular dependence and determine ##\langle \theta, \phi | \ell, m \rangle##, the amplitude for a state of definite angular momentum to be at the angles ##\theta## and ##\phi##. These amplitudes, which are functions of the angles, are called the spherical harmonics and denoted by
$$\langle \theta, \phi | \ell, m \rangle = Y_{\ell, m}(\theta, \phi) \tag{9.132}$$
where
$$\hat L_z\rightarrow\frac\hbar i\frac\partial{\partial
\phi}\tag{9.117}$$ $$\hat L_x\rightarrow\frac\hbar
i\bigg(-\sin\phi\frac\partial{\partial\theta}-\cot\theta\cos\phi\frac\partial{\partial\phi}\bigg)\tag{9.127}$$
$$\hat L_y\rightarrow\frac\hbar
i\bigg(\cos\phi\frac\partial{\partial\theta}-\cot\theta\sin\phi\frac\partial{\partial\phi}\bigg)\tag{9.128}$$
Then he states that since an eigenstate of a hydrogen-like system can be written as ##\vert E,l,m\rangle##, we have
$$\langle r,\theta,\phi\vert E,l,m\rangle=R(r)Y_{\ell, m}(\theta, \phi)$$
Does this mean that he means one could write ##\vert E,l,m\rangle= \vert E\rangle\otimes\vert l,m\rangle## and ##\langle r,\theta,\phi\vert=\langle r\vert\otimes\langle\theta,\phi\vert##?
If yes, then does this mean that when we say we have a simultaneous eigenstate of three variables ##\vert x,y, z\rangle##, what we really mean is that we have a state ##\vert x\rangle\otimes\vert y\rangle\otimes\vert z\rangle##?
One of the most evident features of the position-space representations (9.117), (9.127), and (9.128) of the angular momentum operators is that they depend only on the angles ##\theta## and ##\phi##, not at all on the magnitude ##r## of the position vector. Rotating a position eigenstate changes its direction but not its length. Thus we can isolate the angular dependence and determine ##\langle \theta, \phi | \ell, m \rangle##, the amplitude for a state of definite angular momentum to be at the angles ##\theta## and ##\phi##. These amplitudes, which are functions of the angles, are called the spherical harmonics and denoted by
$$\langle \theta, \phi | \ell, m \rangle = Y_{\ell, m}(\theta, \phi) \tag{9.132}$$
where
$$\hat L_z\rightarrow\frac\hbar i\frac\partial{\partial
\phi}\tag{9.117}$$ $$\hat L_x\rightarrow\frac\hbar
i\bigg(-\sin\phi\frac\partial{\partial\theta}-\cot\theta\cos\phi\frac\partial{\partial\phi}\bigg)\tag{9.127}$$
$$\hat L_y\rightarrow\frac\hbar
i\bigg(\cos\phi\frac\partial{\partial\theta}-\cot\theta\sin\phi\frac\partial{\partial\phi}\bigg)\tag{9.128}$$
Then he states that since an eigenstate of a hydrogen-like system can be written as ##\vert E,l,m\rangle##, we have
$$\langle r,\theta,\phi\vert E,l,m\rangle=R(r)Y_{\ell, m}(\theta, \phi)$$
Does this mean that he means one could write ##\vert E,l,m\rangle= \vert E\rangle\otimes\vert l,m\rangle## and ##\langle r,\theta,\phi\vert=\langle r\vert\otimes\langle\theta,\phi\vert##?
If yes, then does this mean that when we say we have a simultaneous eigenstate of three variables ##\vert x,y, z\rangle##, what we really mean is that we have a state ##\vert x\rangle\otimes\vert y\rangle\otimes\vert z\rangle##?