# Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant.

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115 views

### identify a tensor product by virtue of pure and entangled elements

If I take a tensor product of vector spaces (for simplicity - this could be more general) $V\otimes W$ then of course it is a vector space, but it has additional structure. One way to think about ...
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### Eigenvector of a linear combination of operators is an eigenvector of each operator

Assume $H$ is a Hilbert space and $a_1,\dots,a_n$ are operators with Hermitian adjoints $a_1^*,\dots,a_n^*$, satisfying the canonical commutation relations. Define $N_j=a_j^*a_j$. Assume $v$ is an ...
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### Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
66 views

### Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
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### Does an operator of x commute with the differential operator with respect to x?

While solving a problem in Quantum Mechanics I got an expression $\frac{d}{dx}V(x)-V(x)\frac{d}{dx}$. The first term is just the derivative of the potential but the second one seems a bit weird. Is ...
142 views

### Gradient and Laplacian in $S^1$

I'm trying to solve the particle in a ring problem without embedding the circle in $\Bbb R^3$, by instead taking the entire space to be $S^1$. Unfortunately, I haven't taken differential geometry yet ...
310 views

### Diagonalization of total angular momentum over creation operators for an isotropic harmonic oscillator?

You have an isotropic three dimensional quantum harmonic oscillator so the Hamiltonian is $$H=\frac{p^2}{2}+\frac{r^2}2$$ If you do the creation-annihilation operator-algebra trick and define ...
23 views

### Checking some work on an expectation value problem

I am working on a pretty simple problem (or so it seems it should be) from Griffith's QM text. The problem states: for the probability density function $\rho (x) = Ae^{-\lambda(x-a)^2}$ a) find A b)...
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### Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
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### Finding eigenvalues of a two-state system

Let $$A=\left[\begin{matrix} 2 & -i \\ i & 2 \end{matrix}\right],$$ Show that $U_1 = \dfrac{1}{\sqrt{2}}(\Psi_1+i\Psi_2)$ and $U_2 = \dfrac{1}{\sqrt{2}}(\Psi_1-i\Psi_2)$ are ...
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### Operators in Quantum Theory

Let $U$ be a unitary operator on a Hilbert Space, and let $\phi$ be an eigenvector of $U$ with eigenvalue $\mu$. Show that $|\mu|=1$ ? I know that if $U$ is unitary then $UU^{+}=UU^{-1}=I$ but I'm ...
433 views

### Problem with commutator relations

part a) is fine. part b) is not. A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$.
Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $\rho_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. \$P_{\phi}:=\langle\phi,\,\rangle\...