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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Calculating the Probability of Measuring One Hydrogen Atom

I'm currently enrolled in a statistical mechanics course and am a bit stuck on how to calculate the probabilities of a hydrogen atom in a given state. I'll post the exact question I'm working on and ...
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64 views

Evaluating Matrix Elements of the Density Operator

I'm taking my first course in graduate statistical mechanics and I'm struggling a bit with the math. I think I understand how to use dirac notation, at least the basic stuff for now, but I want to ...
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34 views

Fourier Transform of

Given the quantum state function $\psi_1(x)=\sqrt{\dfrac{2}{a}} \sin{\dfrac{\pi x}{a}}$ , calculate it´s Fourier Trabsform for the momentum, defined as: $$\phi_1(p)=\dfrac{1}{\sqrt{2\pi \hbar}}\int_{-\...
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33 views

Integration of Spherical Harmonics with a Gaussian(Quantum Mechanics)

I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \...
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45 views

Quantum mechanics: First order perturbation to eigenstates

On page 254 of "Introduction to Quantum Mechanics Second Edition", by David J. Griffiths he writes the first order correction $\delta u_n(x)$ to the eigenstates $u_n(x)$ as $$\delta u_n(x)=\sum_{m\ne ...
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1answer
32 views

metastable decay

I am currently dealing with metastable potential in a form of \begin{equation*} V(x) = \begin{cases} \alpha x^2 &\text{ $x\in (-\infty,a] $}\\ -\gamma x &\text{$x>a $} \end{cases} \...
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2answers
105 views

How to compute the levy path integral with zero potential?

In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as $$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,...
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57 views

Why can a Hermitian matrix $A$ be written as $\sum_{j = 1}^{N} \lambda_j|u_j\rangle\langle u_j|$?

I've seen this being used in several quantum mechanics texts but I'm not sure of the reason: Why are we able to write $N\times N$ Hermitian matrices like $A$ having eigenvalues $\lambda_j$ and ...
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28 views

Operator networks, Schrodinger vs Heisenberg picture

We can think of an asynchronous network of operators on a hilbert space to be something like a quantum algorithm. Wires represent the informatic objects, namely quantum states, and nodes represent ...
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37 views

Quantum Mechanics - Orthonormal Basis integration and Kronecker delta

Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...
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40 views

Finding the expansion coefficients of the decomposition of a 3D Gaussian wavepacket in the eigenfunctions of the Hydrogen atom

I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet. First I'm decomposing a Gaussian wave packet $$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/...
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49 views

Finding the correct expression for a square-wave wavefunction with time dependence

First some background information: Eigenstates of the momentum operator with eigenvalue $p$ are given by $$\phi(p,x)=\frac{1}{\sqrt{2\pi \hbar}}\exp\left(\frac{i \,...
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134 views

Dirac delta function like a “function” of an operator

Let $\mathcal{S}\subset \mathcal{H}=L^2(\mathbb{R}^n)\subset \mathcal{S}^*$ be the Schwartz space, the Hilbert space and the space of tempered distributions respectively. Consider an algebra $\mathcal{...
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1answer
48 views

Simplifying Quantum Tensor products with coefficients.

I am trying to show equality of two intermediate steps in the rearrangement of the Quantum Fourier transform definition, but I do not know how to rearrange the coefficients of a tensor product. The ...
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2answers
55 views

Eigenvalues of an operator

I am stuck on this: Deduce the eigenvalues of the operator $A^{\dagger} A$ are positive, where $$A^{\dagger} = -\frac{\text{d}}{\text{d}x} + \tanh(x)$$ $$A = \frac{\text{d}}{\text{d}x} + \...
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Two Dirac delta functions in an integral?

For context, this is from a quantum mechanics lecture in which we were considering continuous eigenvalues of the position operator. Starting with the position eigenvalue equation, $$\hat{x}\,\phi(x_m, ...
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2answers
339 views

What is the difference between vector space and dual space?

I read that in Dirac notation, kets are elements of a vector space and bras are elements of the dual space. My question is, what is the difference between vector space and dual space, and why are bras ...
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3answers
101 views

Can complex vectors be orthogonal even if none of the components are? [closed]

I tried researching this on my own but I couldn't find a concise answer that didn't bog me down with terms and math I don't understand. I'm attempting to read Leonard Susskind's Theoretical Minimum ...
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1answer
48 views

Tricky question involving finding a relation between the time derivative of an operator and a commutator of two operators

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
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74 views

How to find log of “sum of two matrices”?

I want to find log (A + B ) where A and B are matrices. The context is that I want to find the Von Neumann entropy which is given by: $Entropy = - Trace [\rho log (\rho) ]$ where $\rho$ is a matrix....
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52 views

Geometrical Quantization and Connections

Really I think this question boils down to what the physical significance of a connection is. Physically, we can think of a symplectic manifold $(\mathcal{M},\omega)$ as essentially a phase space. ...
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1answer
58 views

Expanding the Square of a Linear Operator - Linear Algebra Question for Quantum Mechanics

I have taken linear algebra and have some experience with basic application to quantum mechanics. I am reading a paper in quantum mechanics and am having trouble following some of the math. Here is a ...
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2answers
92 views

Find the value of the constant $A$ given the particle has the wavefunction $\psi(x,t=0)=A\cos^{3}\left(\frac{\pi x}{2a}\right)$

Consider a particle in an infinite square well with $$V(x) \begin{cases} = 0 & −a \lt x \lt a\\ \to \infty & \text{otherwise} \end{cases}$$ At $t = 0$, the particle has the wavefunction (...
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38 views

How to create an irreducible representation of nonsymmorphic space groups, and go through an example

I want to create irreducible representations of nonsymmorphic space groups, specifically 2D space groups, pg, pmg, pgg, p4g. I've been reading some resources but the way explain them are too abstract ...
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1answer
30 views

Can we see FdHlb as a 2Category of groupoids?

Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos. If so, can we take a ...
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17 views

How can one define the quantum interferometric power in the case of a multiparametric system using the quantum Fisher information matrix?

please, was the quantum interferometric power defined in the case of a multiparametric system? I know that in the case of a single parameter, the quantum interferometric power is defined as the ...
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60 views

Solving a recursive formula

Can somebody please help me solve this? $C_n$ = $2 D^{n-1}$ + $\hat{D}*C_{n-2} * \hat{D}$ where $C_1 = I_{2x2}$ and $C_2 = 2*D $ and D is a matrix operator**. I want to find $C_n$ in terms of D and ...
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53 views

How deep is the connection between Bayesian framework and bra-ket calculus?

As a PhD student in condensed matter physics, I am very familiar with Dirac's bra-ket notation, but not so much with Bayesian inference. One of the first things that struck me when I started studying ...
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1answer
42 views

Decomposition of Schroedinger operator to left and right hand sides

Choose the Hilbert space as $\mathcal{H}:=L^2(\mathbb{R})$ and define on it (or on a closed domain of it) the self-adjoint unbounded Schroedinger operator $H:=-\partial^2+V$ where $\partial$ is just ...
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50 views

How to simplify the time-ordered exponents?

How to simplify time-ordered exponents $e^{iH_1t/\hbar}e^{-iH_2t/\hbar}$? If $H_1$ and $H_2$ presumably don't commute, will $e^{iH_1t/\hbar}e^{-iH_2t/\hbar}$ be presented by a exponential integral?
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47 views

Delta and Theta functions

I have a function $\theta (x-x') $ which is zero when $x-x'$ is negative and one when $x-x'$ is positive. I am to prove, $$\delta (x-x') = d/dx [\theta(x-x')] $$ The following is my reasoning: I ...
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42 views

Find wave function satisfying Schroedinger Equation $[\frac{1}{2m}(\frac{\hbar}{i}\nabla-\frac{e} {c}A(r))^2+V(r)]\psi^{'}(r) = E\psi^{'}(r)$

Given an "ordinary" wave function $\psi(r)$ that satisfies the "ordinary" stationary Schrodinger equation $[-\frac{\hbar^2}{2m}\Delta+V(r)]\psi(r)= E\psi(r)$, I want to construct a wave function $\psi^...
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1answer
84 views

The Dirac delta function

I am to prove that $\delta(ax) =\frac{\delta(x)}{|a|} $ The following is my reasoning: I begin with, $$\int_{-\infty}^{\infty} \delta(x') f(x') dx'=f(0) \tag{1} $$ Where $\delta$ is the Dirac delta ...
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1answer
66 views

Why does this integral vanish while doing integration by parts?

Consider $$\int_{-\infty}^{+\infty} x \dfrac{\partial}{\partial x}\left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right) dx $$ If I apply integraton by parts ...
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1answer
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Phases of the constants $\chi_0=c_1\begin{bmatrix}1 \\ 0\end{bmatrix}+c_2\begin{bmatrix}0 \\ 1\end{bmatrix} \,\,\,\,\, c_1,c_2 \in \mathbb{C}$

I'm considering a vector with complex coefficients. $$\chi_0=c_1\begin{bmatrix}1 \\ 0\end{bmatrix}+c_2\begin{bmatrix}0 \\ 1\end{bmatrix} \,\,\,\,\,\,\,\,\,\, c_1,c_2 \in \mathbb{C}$$ I know that $|...
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48 views

Time-independent Schroedinger equation is first order? Where did I go wrong?

I am attempting to solve the Schroedinger equation for a particle with potential energy: $$V(x) = -Fx$$ The associated Hamiltonian is: $$ H = \frac{p^2}{2m} - Fq $$ In the Schroedinger representation, ...
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1answer
116 views

Prove sum of products of hermitian matrices to be hermitian

A given problem states: If $A$ and $B$ are hermitian matrices, prove that $(AB+BA)$ is hermitian. Because the sum of hermitian matrices is known to be hermitian, the problem seems to me to boil ...
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1answer
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Deriving Rodrigues Formula and Generating function of Hermite Polynomial from $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$

There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, ...
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44 views

How to calculate the integral ${\langle\psi|\overrightarrow{r}|\psi\rangle}=\frac{\alpha^3}{\pi}\int_{V}\overrightarrow{r} e^{-2\alpha r}dV$

I'm considering the following wave function: $$\psi(r)=\sqrt{\frac{\alpha^3}{\pi}}e^{-\alpha r} \,\,\,\,\,\,\, \alpha \in \mathbb{R}$$ Where r is the distance from the origin not the radial vector: $...
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1answer
21 views

Is the inner product distributive in this sense?

Let's say we have $|\langle x|y\rangle|^2=\langle y|x\rangle\langle x|y\rangle$. I have the projection operator $\mathbb{P}=|x\rangle\langle x|$, and a projection operator on the subspace $\mathbb{P}=|...
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1answer
131 views

Is the quantum relative entropy a Bregman divergence?

This is related to a broader question about quantum information geometry - this question is more specific. A fundamental concept in Amari's treatment of information geometry is that of a Bregman ...
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248 views

What is the essential difference between classical and quantum information geometry?

This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory. I have a ...
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1answer
25 views

A general result on scattering off a potential

Consider $V_1$, $V_2$ real potentials, $V1(x) \leq V_2(x) \leq 0$ for all x. $V_i(x) =0$ for $|x|>a$ . A particle obeying 1D shrodinger is input from left to each potential. Is it possible that ...
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1answer
84 views

Arbitrary constants in the general solution to the Schrödinger equation

Are there some general cases for which the costants in the solution of Schrödinger equation for a one dimensional problem are real? I know that in general: $$\psi(x)=Ae^{ikx}+Be^{-ikx} \,\,\,\,\,\,...
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2answers
112 views

Power of $a^{\dagger}=\sqrt{\frac{\omega m }{2 \hbar}}\left (x-\frac{\hbar}{m\omega}\frac{d}{dx} \right)$

I don't know if a mathematical passage is correct. For the moment I am not entirely interested in formally understanding justification, but only if it is correct operatively. Considerig the ground ...
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31 views

Evaluating $|\frac{1}{2}(|a\rangle \otimes|b\rangle+|b\rangle\otimes |a\rangle) |^2$

I have a quantum state of 3-qubits : $|\psi\rangle = \frac{1}{2}|0\rangle (|a\rangle \otimes|b\rangle+|b\rangle\otimes |a\rangle) + \frac{1}{2}|1\rangle (|a\rangle \otimes|b\rangle - |b\rangle\otimes |...
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0answers
39 views

Proving Uniqueness of Momentum Operator

I'm attempting to do two things. One making a well posed question, and two answering that question. Here is the question: I'm trying to show that the operator $- i \frac{\partial }{ \partial x } $ ...
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37 views

Topologically proctected twist in the wave function (Chern number)

In the famous TKNN paper and subsequents the authors wisely argue that the transversal conductance in the Integer Quantum Hall Effect has a topological interpretation as the integral of the curvature ...
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1answer
17 views

Normalizing/integrating a product of two exponentials

In Griffith's Intro to QM Problem 1.5, I'm asked to normalize the wave function: $\psi ( x,t) =Ae^{-\lambda |x|} e^{-i\omega t}$ (where $A$, $\lambda$ and $\omega$ are positive, real constants) I ...
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1answer
169 views

Bloch's theorem in one dimension, confusion about proof

I was looking at the derivation of Bloch's theorem in Griffith's QM: If $V(x+a)=V(x)$ for any $x$ and some $a$, and $\psi$ solves $$ H\psi =\lambda \psi $$ for $H=-\frac{\hbar^2}{2\pi}\frac{d^2}{dx^2}+...