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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2
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1answer
38 views

Distinct ways three integers can sum to a constant

So I am doing some quantum mechanics and it has led to some combinatorics. I need to know how many distinct ways I can have $N_1+N_2+N_3=N$ where $N$ is fixed so we can change $N_1$, $N_2$ and $N_3$. ...
2
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1answer
31 views

Evaluation of Operator-Valued Function

Hello all; above is my question! :) I've gone through all the way up to the final "and hence deduce that". Up to this point, the question has been fairly straightforward, but I have no idea how to ...
2
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1answer
33 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
2
votes
1answer
108 views

Quantum Hamiltonian commuting with the Pauli-Runge vector.

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, ...
2
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1answer
170 views

Left-ratio and right-ratio in (not necessarily commutative) field

I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$. The authors use terms right-ratio and left-ratio in section 13. Right-ratio is denoted as $[x_1, x_2, ... ...
2
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1answer
85 views

boundary conditions for Schr$\ddot{\textrm{o}}$dinger equation in 2D polars?

What are the boundary conditions at $r=0$ for Schr$\ddot{\textrm{o}}$dinger's equation for a quantum particle in 2D polars $(r,\theta)$, with potential $U=0$ for $r<a$ and $U=\infty$ for $r>a$? ...
2
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1answer
122 views

How are Tr(AB) results restricted?

In quantum mechanics one postulates that for each state $i$ there is a matrix $A_i$ and for each measureable dynamical variable $j$ (velocity etc.) there is a matrix $B_j$. Both are Hermitian matrices ...
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0answers
48 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
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0answers
24 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
2
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0answers
38 views

A mixed state integrated over Haar measure in QFT

I think that we can write a state in Quantum Mechanics like: $$ \int dx\lambda _x \vert x\rangle\langle x\vert $$ and when we talk about QFT, we would replace integral with functional integrals: $$ ...
2
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1answer
27 views

Find the constant in the following matrix

An atom is prepared in the angular momentum state $$C\left(\begin{array}{c}1 \\ 2\end{array}\right)$$Here $C$ is a constant. This has benn written in the $S_z$ basis. a)Find C b)Work out ...
2
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1answer
49 views

Application of Fubini Theorem in Quantum Mechanics

I'm afraid I'm very confused by how to correctly apply the Fubini theorem to simplify integrals? I have some integral $$ \sum_{k = 0}^{2}\int_{0}^{T} dt_2 \int_{0}^{t_1} dt_1 \bigg[ ...
2
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0answers
50 views

Proof of Heisenberg Uncertainty Principle Exercise

I'm not very knowledgeable in QM, and I know many physics books derive the uncertainty principle using commutators, but as an exercise in my PDE book (by Asmar), I should be able to derive it from one ...
2
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0answers
88 views

$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
2
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1answer
39 views

How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
2
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1answer
60 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
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0answers
49 views

Invariant set under the flow defined by Schroedinger equations

I have to show that the set of functions of the form $$\psi(x,t)=c(t)^{-1}e^{\frac{-(x-q(t))^2}{2c(t)^2}}e^{ip(t)x}\hspace{1cm}c(t),p(t),q(t)\in\mathbb{R}$$ is invariant (as set) under the flow ...
2
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0answers
49 views

Adding a delta function to a differential equation

So say I have a differential equation of the form: $$ \left(\alpha \frac{d^2}{dx^2}+fx^2 \right)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now ...
2
votes
1answer
42 views

Questions about Commutators

My motivation is understanding some derivations in Quantum Mechanics, but since my question is purely Algebraic, I think it is suited for these forums. I have a general question and then a specific ...
2
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0answers
70 views

group cohomology for SO(3) and SO(3,1)

I am studying relativistic quantum mechanics and I have encountered the concept of projective representation for a group. I have read in http://groupprops.subwiki.org/wiki/Projective_representation ...
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0answers
75 views

Axiom of choice in proof of Wigner's theorem?

In Appendix A of chapter 2 of "The Quantum Theory of Fields," vol. 1, Weinberg presents a proof of Wigner's theorem: given a symmetry transformation $T$ of rays, one can extend this to a symmetry ...
2
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0answers
62 views

Does $Z_A$ exist such that $\exp(X+A) = \exp(X) Z_A$?

I am considering an exponential on the following form: $$\exp(X + A \otimes I_B),$$ where $X$ is a Hermitian operator on a tensor Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, $A$ is a ...
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0answers
75 views

Proving commutation relation in Algebraic Bethe Ansatz

I have a problem with proving a certain commutation relation. For my Bachelor's thesis I give a more mathematically rigurous 'treatment' of a select set of chapters of a paper by L.D. Faddeev. Noting ...
2
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1answer
77 views

Can a unitary matrix be constructed from any doubly stochastic matrix?

Here is a question that came up while I was thinking about the foundations of quantum mechanics: Consider a unitary $n\times n$ complex matrix $U$, with elements $u_{ij}$. We know that the rows and ...
2
votes
1answer
226 views

Finite element method for the 'Particle-In-a-Box' problem in quantum mechanics

(Apologies in advance for the lengthy question, but it really is needed for a precise description of what I've done!) In suitable units, the 'Particle-in-a-box' problem is described by the following ...
2
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0answers
37 views

Expectation value in a Quantum derivation

I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not ...
2
votes
1answer
37 views

Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian.

Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix ...
2
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0answers
68 views

Asymptotic behavior of $L^2$ norm for increased matrix dimensions

I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could ...
2
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0answers
79 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
2
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1answer
34 views

calculating degenerancy

Given a function of two positive integers $n_x^2+n_y^2$. $n_x^2+n_y^2=50$ has three combinations of $n_x$ and $n_y$ that result in $n_x^2+n_y^2=50$: $$n_x=7,n_y=1$$ $$n_x=5,n_y=5$$ $$n_x=1,n_y=7$$ ...
2
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0answers
61 views

Fock Subspaces and Weight Vectors

I've got an assignment due in a few hours, and I'm at a complete loss as to how to even start it, really. I haven't encountered any Dirac notation before, so I'm having a lot of trouble attempting the ...
2
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0answers
43 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
2
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1answer
225 views

1D Schrodinger/Laplace equation via finite differences: incompatible eigenvalues

I need to solve a variant of the 1D Schrodinger's equation equation using finite differences, so I decided to play a little bit with the real-space representation of some operators. Using the ...
2
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0answers
54 views

To construct a Schrödinger wave with prescribed mean position and momentum

Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
2
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0answers
312 views

Is quantum game theory reducible to classical game theory? [closed]

Modnote: This question was manually migrated (closed and crossposted) to MathOverflow by request of the OP. Quantum game theory is an extension of classical game theory to the quantum domain. It ...
2
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0answers
112 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
2
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0answers
52 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 $A$ ...
2
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0answers
134 views

How do I solve this integral

How do I solve this integral (expectation value) : $$\int_{-\infty}^{\infty} \psi (x)^* \hat p \psi (x)\ dx.$$ where the $\hat p =-i\hbar \frac {\partial}{\partial x}$ is an operator and $\psi (x)$ is ...
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4answers
102 views

Solution to a system of quadratics

I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system $$\alpha_0^2+\alpha_1^2=1$$ $$\beta_0^2+\beta_1^2=1$$ ...
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2answers
78 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
1
vote
1answer
93 views

Commutator proof

I have a proof in my book I dont fully understand. The author is proving that if $[A,B]=1$ then $[A,B^n]=nB^{n-1}$. The proof is really short, it is only one line of equations, but I dont understand ...
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vote
2answers
200 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
1
vote
1answer
111 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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2answers
88 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.
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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 ...
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vote
2answers
189 views

Under What Conditions Does the Action of the Dual Space Induce an Hermitian Inner Product?

I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space ...
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3answers
57 views

looking for help with a trace/norm inequality

I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
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2answers
52 views

Creating a Hermitian function

Say I have an operator $A$ such that $A^\dagger = B$. I want to construct a Hermitian function, $f$, of these operators, $f(A,B)^\dagger = f(A,B)$. Is it possible to construct a function $f$ such that ...
1
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3answers
53 views

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 ...
1
vote
1answer
65 views

Fourier transformations in Simon's quantum paper

I am reading this paper by Simon. This is one of the earliest quantum algorithm papers. In the paper he presents a routine starting at the end of page six. The first step runs a Fourier transformation ...