Linked Questions
12 questions linked to/from How to construct the radial component of the momentum operator?
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Hamiltonian operator in polar coordinates with momentum operators
The Hamiltonian operator for a free non-relativistic particle looks like
$$
\hat{H} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m} \nabla^2. $$
In polar coordinates, the Laplacian expands to
$$
\...
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answer
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Derivation of the radial momentum operator
I have been studying Quantum Mechanics and when my book was going through the Hydrogen wave equation, it was talking about this equation:
$$ \frac{p_r^2}{2\mu} +\frac{L^2}{2\mu r^2}+V(r)=E$$
I ...
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answers
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Imagining zero orbital angular momentum for s-orbitals
Orbital Angular momentum of a s-orbital is always zero. One can easily imagine why this is so: QM says $\hat{p}=-i\hbar \nabla_{r}$, and since the s-wave functions are radially symmetric, the momentum ...
- 161
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Expected momentum of ground state hydrogen $<p>$
I am trying to calculate the expected momentum of an electron in the ground state of hydrogen atom. This is the wave function.
So far I have done this:$$\iiint_V \Psi^* (-i\hbar) \frac {d\Psi} {dr} ...
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How to prove radial momentum operator is hermitian?
Technically, if we want to prove an operator $\hat{A}$ to be an hermitian,we should prove $
\left< \psi \hat{A}|\psi \right> =\left< \psi |\hat{A}\psi \right>$.
It works well in Cartesian ...
- 171
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Clarification in deriving the radial momentum operator $p_r$
In deriving an expression for $p_r$, a particle's radial momentum, I am unsure what is happening at a certain step. The derivation given in The Physics of Quantum Mechanics by Binney and Skinner is as ...
- 442
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Momentum of a hydrogen system is imaginary. Why?
The hydrogen ground eigenstate:
$$
\psi(r) = \frac{1}{\sqrt{\pi r_0^3}}\exp\left(\frac{-r}{r_0}\right)
$$
Notice how nice:
$$
\frac{\partial \psi}{\partial r}(r) = \frac{1}{\sqrt{\pi r_0^3}}\frac{-1}{...
- 3,334
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1
answer
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Canonical commutation relation for spherical coordinates?
What coordinates systems can the canonical commutation relation be generalised to? I also ask specifically for spherical coordinates. This is because I want to prove $\hat{\bf p}\cdot{\bf e}_r-{\bf e}...
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Is the radial component of $p^2$ a hermitian operator?
The kinetic energy of a system $p^2/2m$ in real space representation takes the form $-\hbar^2\nabla^2/2m$. I want to express this in cylindrical coordinates via the representation of the laplace ...
- 828
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Momentum operator in spherical coordinates
In the position representation the momentum operator takes the form of the gradient, $-i\hbar \nabla$.
It is understood that its components denote $p_{x}, p_{y}, p_{z}$ respectively; but, when ...
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2
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Hermiticity of a radial momentum operator $\hat{p}_r$ and the spectral theorem
In Nolting's QM book (Theoretical Physics 7), in the chapter on central potentials, a radial momentum operator $\hat{p}_r$ is defined as
\begin{equation}
\hat{p_r} = -i \hbar \Big( \frac{\partial}{\...
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How do we determine the conjugate of radial component of linear momentum?
How do we describe the radial part of linear momentum?
Here I found the following description:
Classically
$$p_r=\hat{D}_r = \frac{\hat{r}}{r} \cdot\hat{p} = \frac{\hbar}{i}\frac{\partial}{\partial r}...
