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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$C^*$-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) ...
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1answer
59 views

Position Operator on $l^2(\mathbb{Z})$

I'm very familiar with the position operator $(Q\varphi)(x)=x\varphi(x)$ on $L^2(\mathbb{R^d})$, but I'm trying to figure out how to interpret the same operator on $l^2(\mathbb{Z})$ (the space of ...
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127 views

Separation of variables and quantum mechanics

In the book Quantum mechanics by Eugen Merzbacher, third edition, at page 462 he claims that this differential equation (for the unknown operator $F_0=F_0(x,y,z)$) can be solved by separation of ...
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1answer
128 views

Probability and Quantum mechanics

I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...
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2answers
222 views

What is the meaning of “Hermitian”?

Google search-bar gives the definition of Hermitian as: Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal ...
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1answer
141 views

Computing a Projection Valued Measure

I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...
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1answer
38 views

Toffoli gates can be decomposed into single and two-qubit gates

I'm not sure what the "I" and "-I" gates do. I can't seem to apply them correctly. When I do hadimard I get |00>(Tensor)Hadimard. If I then apply the tensor product to apply the 'i' gate on the last ...
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63 views

Two Body Schrodinger Equations

I have a question involving the eigenvalues of a two-body Schrodinger equation. Let $$H=-\frac{1}{2m}\Delta_{x_1}-\frac{1}{2m}\Delta_{x_2}+\frac{e^2}{|{{x_1}-{x_2}}|}$$ over the Hilbert space ...
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1answer
24 views

Regarding matrix representation of $SO(4)$

As per the title of the question, what are the matrix elements of the special orthogonal group $SO(4)$? I'm not certain but I believe they are somehow related to being operators of angular momentum ...
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3answers
87 views

Addition in linear vector spaces

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can ...
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1answer
45 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 $\hat\rho$, ...
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88 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
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46 views

Why such strange separation of variables?

In this article the authors give the following expansion of wavefunction of three-body system (equation $(16)$ in text): $$\Psi(\textbf{x},\textbf{y})=\sum_{q=\lambda}^l\psi_q(\textbf x^2,\textbf ...
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1answer
95 views

Quantum fourier transformation Unitary proof.

I've found a bunch of these proofs online but I am having trouble understanding how the norm of the column/row is 1.
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31 views

Matrices with Continuous Indices

The components of a matrix $A$ can be written as $a_{ij}$. In Quantum we're starting to talk about a generalization where the indices are not elements of $\Bbb N$, but are instead continuous. Our ...
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18 views

Rotation in configuration space.

Let $R_\psi$ be the rotation in configuration space around a vector $\bf{e}_\psi$ for an angle $\psi$. How is that the space rotation in configuration space have: ...
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1answer
71 views

How to derive the hamiltonian from a non-classical lagrangian

For the non-classical lagrangian of a hydrogen atom: $$L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e^2}{4 \pi \epsilon r}$$ We get that two conserved quantities are: $J = \gamma mr^2 \dot{\phi}$ and ...
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1answer
32 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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36 views

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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1answer
48 views

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 ...
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1answer
41 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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1answer
54 views

Spectrum of Rank 1 Operators

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and ...
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38 views

Momentum Representation vs Position Representation

I have a question involving the representation of operators in momentum representation and position representation. The question is a little long, so I'll do my best to explain it. We are given an ...
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1answer
31 views

Can a general time-dependent finite-dimensional Schrödinger equation with complex Hamiltonian be transformed to one with real Hamiltonian?

Consider a general-form time-dependent Schrödinger equation: $$i\partial_tv=\hat Hv,$$ where Hamiltonian $\hat H$ is an Hermitian matrix (finite-dimensional for simplicity), and $v(t)$ is a complex ...
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1answer
28 views

When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary Ĥ and I want to know, if Û is some other unitary, when is Ĥ Û a Hermitian unitary? Specifically, what are the conditions on Û such that ...
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3answers
59 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 ...
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1answer
118 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 ...
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86 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 ...
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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 ...
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2answers
94 views

Question about Dirac notation

So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p ...
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1answer
62 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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110 views

Quantum mechanics, conmutative operators.

If two operators $A$ and $B$ commute then any eigenvector of $A$ is an eigenvector of $B$? I know that if that happens there is a basis in which the eigenvectors of $A$ and $B$ are equal, but I don't ...
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3answers
84 views

How to do this integral $\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$ [duplicate]

How to do this integral $$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$ for any $k > 0$ ?. I tried to use gamma function, but sometimes the series doesn't converge.
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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 ...
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1answer
23 views

Sign of energy and solving the Schrodinger equation.

The particular problem that triggered my question is as follows: A particle of mass m is confined within the box $0 < x < a$, $0 < y < a$ and $0 < z < c$. The potential vanishes ...
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73 views

Understanding the Quantum Fourier Transform

I have a question about the Quantum Fourier Transform. I would like to understand it because I have a re-take for an exam. I have studied the provided / recommended literature extensively. ...
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0answers
56 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 ...
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Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute ...
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1answer
32 views

Wavefunction of electron above grounded conductor

Consider a non-relativistic electron moving above a large, flat grounded conductor while it is attracted by its image charge, but cannot penetrate the conductor's surface. What is the Hamiltonian of ...
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2answers
513 views

A vector space with countable and uncountable basis at the same time

Let $V$ be a vector space over $\mathbb{C}$. Two self-adjoint, commutable linear operators $\xi$ and $\eta$ act on it. Both of their eigenvectors form a complete set of $V$, but $\xi$'s eigenvalues ...
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39 views

Square Root of matrices and multiplication

In David Tong's Quantum field theory lecture notes, page 101, line 5, he shows that: $$(p.\sigma)(p.\overline{\sigma})=m^2.I_2$$ (I have placed the identity matrix $I_2$ for clarity) Then I don't ...
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1answer
161 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
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1answer
77 views

Determining Degeneracies of Operator (Quantum Mechanics / Linear Algebra)

Hello all! Above is my question. I am fine all the way up to the final part about the degeneracy. I find counting degeneracies quite difficult, and this is no exception! I have really no idea why ...
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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 ...
3
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1answer
145 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
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1answer
68 views

What is the smallest non-trivial Hilbert space?

I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof? One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
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1answer
1k views

How to show Thomas-Reiche-Kuhn sum rule

Given: $$ f_{ni} = \frac{2m\omega_{ni}}{\hbar}\big|\langle n | x |i\rangle\big|^2, $$ and a Hamiltonian in the form $$ H_0=\frac{p^2}{2m}+V(x), $$ I would like to show the following sum rule (known as ...
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1answer
28 views

Re-expressing the Schrodinger Equation as a first order expansion.

I am reading an online text on quantum computing and the author expands and re-expresses the Schrodinger equation. I am not really sure as to the intermediate steps he used or what happened to the ...
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2answers
66 views

Constructing 5 by 5 Unitary matrices

I am trying to construct an arbitrary 5 x 5 Unitary matrix. Any example will be appreciated.
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1answer
43 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 ...