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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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2answers
117 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle \...
2
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0answers
78 views

Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
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0answers
53 views

Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials). Is ...
2
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1answer
536 views

Tensor and Kronecker product

I have a follow exercise. Let $\left|\Psi\right\rangle = \tfrac{1}{\sqrt{2}}\left(\left|0\right\rangle + \left|1\right\rangle\right)$. Write out $\left|\Psi\right\rangle^{\otimes 2}$ xplicitly, both ...
2
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1answer
132 views

Translation of an article

I need to read this article "On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups" authors:G.Zhislin, A. ...
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0answers
32 views

References for three body problems with Fermi statistic

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
3
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1answer
336 views

Is the partial trace congruent under a change of basis?

My intuition tells me that the partial trace should be congruent under a change of basis. That is, if I have some matrix $A$ in the space of linear operators acting on a joint hilbert space: $A \in \...
0
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1answer
87 views

Dirac's notation? (QM)

I have a question regarding Dirac's notation in quantum physics. As far as I understand: $\langle a|b\rangle=(a1^*,a2^*)*(b1,b2)^T$ But what does $\langle1/2,1/2|J|1/2,-1/2\rangle$ mean?
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0answers
206 views

Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is $$K(x,y)...
4
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1answer
381 views

General theory behind ladder operators

To derive the representation of SO(3) one uses the ladder operator method. What is the theoretical basis for this method? Often the ladder operators are simply stated in the textbooks of quantum ...
-1
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1answer
131 views

Qubit state finding [closed]

Suppose we have two qubits in the state $x|00\rangle+y|11\rangle $. What is the resulting state of the second qubit in that case? Use and to denote and respectively.
2
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1answer
264 views

Is multiplying by a measurable function $V$ always self-adjoint?

There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator: $$-\Delta + V(x)$$ Is (essentially) self-adjoint on $L^2(\mathbb{R}^n)$....
3
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1answer
242 views

A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
2
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1answer
246 views

Question on complex number calculation for transmission coefficient of finite potential well

This is actually in my quantum mechanics textbook (pure math question though), and I just cannot see why this equality is true. Any help would be greatly appreciated! Let $F$ and $A$ be nonzero ...
6
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1answer
620 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
4
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1answer
48 views

Essential selfadjointness preserved under unitarily transfomration?

I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations. In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an ...
4
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5answers
148 views

General solution of $\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\rho^{2}}u$

In his discussion of the radial wave function of hydrogen Griffiths (Introduction to Quantum Mechanics, 2nd ed, p.146) gives the general solution of$$\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\...
1
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2answers
92 views

Show $e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$

How do you show that $$e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$$ where $\sigma_i$ are the Pauli matrices.
3
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1answer
546 views

Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?

Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$ H = H_1 \otimes \cdots \otimes H_n, $$ and let $\mathcal{H}$ be a ...
8
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1answer
185 views

A non-integrable representation of the Heisenberg Algebra

Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ ...
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0answers
618 views

Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?

This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
1
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0answers
121 views

Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (https://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
2
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2answers
999 views

Solving the time-independent Schrodinger equation for particle in a potential well

I'm solving a quantum mechanics problem for the particle in a potential well, and the equation I have to solve is $$\frac{d^2\psi}{dx^2}+k\psi=0$$where $$k=\frac{2mE}{\hbar^2}$$ This seems easy enough ...
8
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2answers
484 views

“Fun” question: anyone know why $e$ (Euler's Number) was chosen for wave functions?

First, let me say that this is merely something I have always wondered about, and can never seem to find a good reference for. I simply want to know... the geek in me. Why was $e$ (Euler's Number) ...
1
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1answer
50 views

Manipulating derivatives after substitution: $\xi=\gamma x$

I am following a quantum mechanics text book which uses a simple looking substitution in a derivative. The substitution is $$\xi=\gamma x\tag1$$ It then says that $$\frac{d\psi}{dx}=\frac{d\psi}{d\...
2
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2answers
117 views

Is there a reason for the similarity between $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$

I was wondering whether the similarity between the functions $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$ was due to some more fundamental limiting relation between the two functions (or similar ...
2
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2answers
95 views

Showing that $\langle p\rangle=\int\limits_{-\infty}^{+\infty}p |a(p)|^2 dp$

How do I show that $$\int \limits_{-\infty}^{+\infty} \Psi^* \left(-i\hbar\frac{\partial \Psi}{\partial x} \right)dx=\int \limits_{-\infty}^{+\infty} p \left|a(p)\right|^2dp\tag1$$ given that $$\Psi(...
1
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1answer
48 views

Unwanted $i$ floating around when trying to calculate $\langle p\rangle$

$\def\sp#1{\left\langle#1\right\rangle}$I am given $$ \Psi(x,0)=A_0 \exp\left(-\frac{x^2}{2\sigma_0^2}\right) \cdot \exp\left(\frac{i}{\hbar}p_0x\right)\tag1$$ where $A_0=(\pi \sigma_0^2)^{-\frac{1}{...
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0answers
736 views

The general recipe for finding the conjugate of a complex function

I have the general recipe for finding the complex conjugate of a function down as follows: Suppose I have $f(z)$: Separate $f(z)$ into a sum of real and imaginary functions such that $$f(z)=u(x,y)+...
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1answer
2k views

Solving time dependant Schrodinger equation in matrix form

If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t),)$$ with Hamiltonian $H$ given by $$H=\hbar\omega \begin{...
1
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1answer
334 views

Two-particle operator in the second quantization

In "Quantum mechanics" by Schwabl I found a chapter (1.3.3) about one- and two-particle operators in the second quantization. The derivation was only sketched and contained this equation: $\sum_{\...
1
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1answer
2k views

Show that $A^{\dagger^{\dagger}} = A $

How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that $\...
2
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3answers
497 views

Instance of Ehrenfest's Theorem

Please Help me to fill in the gaps Show $$ \frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$ $$\frac{\text d \langle {p} \rangle}{ \...
8
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1answer
3k views

Relationship between dual space and adjoint of a linear operator

I am having a hard time understanding the concept of adjoint of a linear operator. Given a finite dimensional Hilbert space $H$ over a field $F$, I know the dual space is the vector space $H^*$ of all ...
1
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1answer
149 views

Eigenstate and quantum mechanics position opperator

Quantum mechanics math question: Suppose that there is eigenstate $|q \rangle$ where $q$ is position observable . The question is, 1) What is eigenstate? How is this different from eigenvector? ...
1
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1answer
3k views

Commutator relationship proof $[A,B^2] = 2B[A,B]$

I'm trying to find the condition necessary for this commutator relationship equality: $$[A,B^2]=2B[A,B]$$ So far I've done this: \begin{align*} [A,B^2] & = B[A,B] + [A,B]B \\ &...
3
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2answers
164 views

Mathematical explanation of problems behind time and space derivatives being second order

$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$ with the wave function $\phi$ being a relativistic scalar: a complex number which has the ...
1
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1answer
94 views

substitution in a non linear differential equation and to get a nicer form

well I had this equation at the begining $$ i \frac{\partial u}{\partial{z}} + \frac{1}{2 k_0} \frac{\partial^2 u}{\partial x^2} +\frac{1}{2}k_0 n_1 F(z) x^2 u-\frac{i[g(z) -\alpha(z)]}{2}u + k_0 n_2|...
2
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2answers
3k views

“proof” A is a Hermitian Matrix

For an arbitrary complex matrix A show that $$A*A^\dagger$$ is Hermitian. Where the dagger "$\dagger$" stands for the "complex conjugate and transpose" operators. From what I understand this must ...
3
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5answers
1k views

Hydrogen atom in partial differential equations

For the hydrogen atom, if $$\int |u|^2 ~dx = 1,$$ at $t = 0$, I am trying to show that this is true at all later times. What I need help is with differentiating the integral with respect to $t$, and ...
2
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0answers
147 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where $\mathcal{H}=\mathcal{H}_1\...
5
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1answer
187 views

Quantum Information: Deutsch-Jozsa Algorithm

There is a step in the construction of this algorithm which I'm not understanding: $\displaystyle \left[\sum_x \frac{| x \rangle}{\sqrt{2^n}}\right]\left[\frac{ | 0 \rangle -| 1 \rangle }{\sqrt{2}}...
7
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1answer
258 views

'Quantum' approach to classical probability

Quantum mechanics defines a state of a system as a superposition of 'classical' states with complex coefficients, thus reducing many problems to linear algebra. Can classical statistics be approached ...
3
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1answer
877 views

What are the requirements for a “test” function to show operators commute?

To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$. My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...
7
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5answers
1k views

Applications of Operator Algebras to modern physics

I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
0
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1answer
293 views

Partial trace of a system with isolated evolution

Let $\rho_{AB}$ be the state of a composite quantum system with state space $H_A\otimes H_B$ (two finite dimensional Hilbert spaces). Now assume that $A$ and $B$ are isolated and suffer a unitary ...
0
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3answers
1k views

Re-writing in sign basis.

$\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha_0$? I came ...
1
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1answer
2k views

Qubits and vector projections

In $\Bbb C^2$, how many real unit vectors are there whose projection onto $|1\rangle$ has length $\sqrt{3}/2$? I would think zero as $\bigl(\frac{\sqrt{3}}{2}\bigr)^2 + x^2 = 1$, therefore there are ...
11
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1answer
537 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
2
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2answers
377 views

Perturbation theorem of Weyl

Does anyone know where to find something about the perturbation theorem of Weyl, preferably on the internet. The theorem I'm talking about states: let $A$ be a self-adjoint operator on a Hilbert ...