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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What geometry or topology best embodies the nonlocality of quantum entanglement?

I am a Princeton physics major. What geometry or topology best embodies the nonlocality of quantum entanglement? https://en.wikipedia.org/wiki/Quantum_entanglement: "Each particle cannot be ...
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Derivative $\Lambda(\phi,\psi)=\frac{\langle\phi|A|\psi\rangle}{\langle\phi|\psi\rangle}$

I' m so sorry I found this expression on a handwritten sheet so I would like to check that it has sense and exactly what it means because I have not found a similar expression on any book. Let the ...
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71 views

Schrodinger equation $iψ'(t) = H(t)ψ(t),\,\,\,\,\,\,\,\,\,\, ψ(t_0) = ψ_0,$

(Quantum Mechanics). A quantum mechanical system which can only attain finitely many states is described by a complex-valued vector $ψ(t) ∈ \mathbb{C}^n$. The square of the absolute values of the ...
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Which Area of Math Contributes to Quantum Theory

Hello Math Community, Thank you for taking the time to read my question. It is much appreciated. I'm curious as to which branch of mathematics would help develop our understanding of quantum theory ...
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101 views

Schrödinger Equation- ODE [closed]

Problem 3.29 from Gerald Teschl ODE. Let the Schrödinger Equation, $$i\psi'(t)=H(t)\psi(t),\ \psi(t_{0})=\psi_{0},$$ where $H(t)$, is a self-adjoint matrix, that is, $H(t)^{*}=H(t) $ . Show that the ...
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129 views

Rotation and Translation Operators

Let $T_y(a)$ be a translation operator of a displacement $a$ parallel to the y-axis. In other words, $$T_y(a)\vec{r}=\vec{r}+a\;\hat{y}$$ If $R_x(\theta)$ is a rotation of $\theta$ around the x-...
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$H_{SO}=f(r)\overrightarrow{S}\cdot\overrightarrow{L} \,\,\,$ $[H_{SO},L_z] \neq 0$

Let $$H_{SO}=f(r)\overrightarrow{S}\cdot\overrightarrow{L}$$ I know that: $$[H_{SO},L_z] \neq 0$$ $$[H_{SO},S_z]\neq 0 $$ but I can not get or find the precise result.
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Please prove the statement in description given in section 3 of Quantum Mechanics (Landau & Lifshitz).

$$ (\;\int_{-\infty}^{\infty} \left |\Psi_n(q)\right |^2dq=1 \; \forall n \;)\;\;\&\;\; (\;\Psi(q) =\;\sum a_n\Psi_n(q)\;) \implies\; \sum \left |a_n\right |^2= \int_{-\infty}^{\infty} \left |\...
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Convergence of delta function in $L^{\varepsilon + 1/2}$

I am currenly reading a paper which states (but does not prove) that any sequence of delta functions converges to zero in $L^{\varepsilon + 1/2}$ for $\varepsilon < 1/2$. Is there a good reference ...
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Mistake in calculation $|c_l|^2\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta \sin(\theta)(\sin(\theta))^{2l}=1$

I did probably a mistake in calculation but I cannot find it. I start from $$|c_l|^2\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta \sin(\theta)(\sin(\theta))^{2l}=1$$ $$I_{l}=\int_{0}^{\pi}d\theta \sin(\...
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Reduced density operators of a pure Bipartite state?

In (Bellucci, 2010; pg89) it is said (my wording): A pure bipartite state is separable if and only if the two reduced density matrices are pure. Proving the "only if" part is easy but I am ...
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32 views

Find a Unitary transformation from $(\phi_n(x)=\frac{1}{\sqrt{a}}e^{i\frac{2\pi nx}{a}})_{n=0, \pm 1,…}$ to trigonometric basis in $L^2([0,a])$

I want to prove that the trigonometric functions form a complete orthonormal system for the space of square-integrable functions on the interval $[0,a]$, a.k.a. that $$ \psi_0 =\frac{1}{\sqrt{a}},\...
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Prove that complex exponentials $(\phi_n(x)=\frac{1}{\sqrt{a}}e^{i\frac{2\pi nx}{a}})_{n=0, \pm 1, \pm2,…}$ are complete and orthonormal on $[0,a]$

I want to prove that the complex exponentials $(\phi_n(x)=\frac{1}{\sqrt{a}}e^{i\frac{2\pi nx}{a}})_{n=0, \pm 1, \pm2,...}$ form a complete orthonormal system for the space of square-integrable ...
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50 views

Problem in quantum information theory

I am reading the following paper: on quantum information theory Does anybody understand the estimate at the bottom of page $7$ $$\left\lVert \rho-\rho_n \right\rVert \le 2 \left\lVert (id_A-P_n)\...
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Number theory to estimate lower bound of spectrum in quantum mechanics?

I recently worked on the following idea: Eigenvalue of an Euler product type operator? Summary of the idea We represent numbers by infinite dimensional matrices such as $3$ will have all $0$s except ...
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Closed form/ meaning of sum of geometric series of operator exponentials

I'm taking my first class on quantum mechanics right now and we've been using various operators throughout it; one operator we derived was the displacement operator, $$e^{a\frac{d}{dx}}f(x)=f(x+a)$$ I ...
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52 views

Bogoliubov Transform

For the operator defined as polynomial is the boson creation and annihilation operators $\hat{a}$, $\hat{a}^\dagger$ such that $[\hat{a},\hat{a}^\dagger] = 1$ $$\hat{L} = A\hat{a}^2 + B\hat{a}^{\...
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Geometric Quantization and Wigner Functions

Geometric quantization of Kähler manifolds successfully leads to the Segal-Bargmann, or holomorphic, representation. The polarized sections naturally lead to a distribution weight given by the Husimi ...
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90 views

Are all the unitary irreducible representations of $SU(2)$ have unit determinant?

The unitary irreducible representations of the group ${\rm SU(2)}$ are $(2j+1)$ dimensional where $j=0,1/2,1,2,3/2...$etc. Consider such a representation of dimension $3$ which corresponds to $j=1$. ...
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29 views

Schrödinger equation energy level

Currently going through an old question which concerns a particle of mass $m$ on the interval $[-a,a]$ with potential $V=V_0$. In the question it says to show that the energy levels of the particle ...
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112 views

Radial probability density inside a hydrogen atom

If the wavefunction describing the state of a hydrogen atom can be written as the product of two functions $R(r)$ and $Y(\theta, \phi)$: $\psi_{nlm} = R_{nl}Y_{lm}$ then the probability density ...
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174 views

Can the quantum harmonic oscillator be solved by power series methods without going for asymptotic analysis?

Although this is a question pertaining to Physics, since this is related to the mathematical treatment of a differential equation, I believe it is well suited for this community. While deriving the ...
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155 views

Surface integral of the Gaussian curvature covering the vertex of a cone

The Gaussian curvature of a cone is undefinable at the vertex, and vanishes elsewhere on the cone. However, the cone is an ideal surface which must not be excluded from quantum mechanics where the ...
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QFT, more a QM Question, Hamiltonian relation time evolution

The problem statement, all variables and given/known data Question attached here: I am just stuck on the first bit. I have done the second bit and that is fine. This is a quantum field theory course ...
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Computing Complex Ito calculus for stochastic process

Let $X_t$ be a stochastic process in $\mathbb{C}^n$ such that $$ dX_t = a(X_t,t)dt + b(X_t,t)dW_t.$$ And let $f:\mathbb{C}^n \to \mathbb{C}$. Then how to compute $df(X_t)$ in complex? If $f$ is a ...
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60 views

What h.c. stands for?

I came across the following equation $$ A = UAV + h.c. $$ For example, please see Eqn (2) in here. But I have no idea what h.c. stands for... It seems that it comes from some physics. Any comments ...
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An operator bound on some state and the state operator

Thanks in advance. I'm struggling with some statement in a paper. It claims as follows, For a p.s.d. operator $H:\mathbb{R}^d\to\mathbb{R}^d$ with an inverse and some $v\in\mathbb{R}^d$, such that ...
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126 views

How to find eigenvalue of $\sigma$ and $\sigma^2$

How to find $\sigma^2$ for $\sigma^2= \sigma_x^2 + \sigma_y^2 + \sigma_z^2$ where $\sigma_x^2$, $\sigma_y^2$ and $\sigma_z^2$ are the identity matrix? My attempt: $\sigma^2 = 3 {\bf I}$ (identity ...
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On common eigenbases of commuting operators

If $A$ and $B$ are self-adjoint operators, each of which possesses a complete set of eigenvectors, then $AB = BA$ if and only if there exists a complete set of eigenvectors which are eigenvectors of ...
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81 views

Real eigenvalues of continuum spectrum of a self-adjoint operator

Is my understanding that if you assume eigenvectors of a self-adjoint operator are in Hilbert space, then is easy to prove that the eigenvalues must be real. However, it could happen that such ...
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The Density Matrix $\rho$ and the Von Neumann Entropy

We know that the Von Neumann entropy can be given by the following formula (up to constants): $S=-\text{Tr}(\rho \log(\rho))$ Where $\rho$ is the density matrix. Now for situations such as ...
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Physically Meaningful Solutions [closed]

I have this in my notes where: $$\Phi(x) = \left\{ \begin{array}{ll} \Phi_I(x) = Ae^{k^{\prime}x} + Be^{-k^{\prime}x} &\text{if }x<0\\ \...
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Normalisation of a free particle with Gaussian wave packet

The Gaussian wave packet where the $x$-dependence is given by the wave function $$\Phi(x) = N\exp\bigg(ikx - \frac{x^2}{2\Delta^2}\bigg)$$ $N$ is a normalisation constant. $k$ is the wave number. ...
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132 views

Proof of identity using polynomial operator and commutators

Suppose x is a real number, and $\hat{A}$ and $\hat{B}$ are two non-commuting operators. A polynomial operator is defined by $$g(\hat{B}) = \sum_{n=0}^\infty a_n \hat{B}^n,$$ where $a_n$ are real ...
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51 views

Perturbation of eigenvalues of Schrodinger Operator.

Suppose I have a sequence of Schrodinger opertors $$ T_n=-\Delta +V_n $$ acting on (a subdomain of) $L^2(\mathbb{R}^d)$. Suppose that I view them as perturbations of the operator $$ T=-\Delta+V $$ ...
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Non-standard Generalizations of the Pauli Matrices that retain closedness, identity mapping & tracelessness?

The Pauli matrices $\{\sigma_{0}=I,\sigma_{1}=X,\sigma_{2}=Y,\sigma_{3}=Z\}$ exhibit the following properties; $$ \sigma_{i}\sigma_{j}=\pm \mathrm{i} \sigma_{k} $$ $$ \sigma_{i}^{2}=\sigma_{0}=I $$ $$...
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Does a linear transformation/function imply an information-conserving transformation?

I've been studying quantum mechanics recently, which is basically just a lot of linear algebra. There's a standard theorem called the no-cloning theorem (and related no-deleting theorem) that says you ...
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57 views

Properties of the monad for the LieAlgebra adjunction

The Lie Algebra is indeed an algebra for a monad. We see some background here. As such, the category of Lie Algebras has an adjunction into Set. This means that there is a monad, $(L,\mu, \eta)$ on ...
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Mathematically rigorous statement and proof of Ehrenfest Theorem

In my quantum mechanics class we proved the Ehrenfest theorem for a conservative physical system with time-independent potential $V:\mathbb{R}^N\to \mathbb{R}$. The issue is that many steps in the ...
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Hilbert space theory and their applications

What are the best books that discuss the theory of Hilbert spaces and their applications to quantum mechanics for the beginners
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318 views

Transition Probabilities Between Two States

I'm currently working on the following problem for my Quantum Mechanics course, I could use some help with one of the integrals involved. A particle of mass $m$ and charge $q$ is confined in a one-...
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Algebra proof for quantum mechanics ladder operator

My question contains the word quantum mechanics but is a purely algebraic problem. The so-called ladder operators $a_{-}$ and $a_{+}$ from quantum mechanics are operators that do not commute and the ...
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General solution to initial value problem of free particle for 1D Schrodinger equation

I was reading Griffith's book Introduction to Quantum mechanics and found that for the case of a free particle, we can diregard solutions of the form $e^{kx}$, where $k$ is real (positive or negative)....
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Question concerning Stirling’s Approximation

From Nielsen & Chuang, page 55: The basic intuition for this decrease in resources required can be understood quite easily. Suppose the source emitting states $|0\rangle$ with probability $p$ ...
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30 views

Potential step Wave function

There is a potential $U(x)$ defined by : $$U(x) = \begin{cases} U_1 & \text{if $x>0$} \\[2ex] U_2, & \text{if $x<0$} \end{cases}$$ such that $U_2>U_1$. The Schrodinger equation ...
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134 views

From the condition $[A,B]=A$, what can I say about $B$?

I'm struggling in understanding the meaning of this condition that I found in an operator equation: \begin{equation} [A,B]=A \end{equation} where both $A$ and $B$ are hermitian operators. What can I ...
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59 views

$a^{\dagger}a(a^{\dagger})^{n-1}+(a^{\dagger})^{n-1}=n(a^{\dagger})^{n-1}$ [closed]

Let $a$ and $a^{\dagger}$ be annihilation and creation operators. I can't prove the following: $$a^{\dagger}a(a^{\dagger})^{n-1}+(a^{\dagger})^{n-1}=n(a^{\dagger})^{n-1}$$ Maybe it's silly, in that ...
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216 views

Quantum Chemistry book recommendation. [closed]

I am trying to learn quantum chemistry. I have an extensive background in math and physics, so I'm looking for a book that makes full use of whatever physics and mathematics is relevant to this ...
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Quantum mechanics continuous index and $ | E , \delta E \rangle = \frac{1}{\sqrt{\delta E}} \int_{E}^{E + \delta E} dE' |E' \rangle$

I'm studying quantum mechanics and my professor has not made a full discussion on vectors with continuous index: the textbook that I follow is Cohen. However he introduced some notions and ...
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54 views

Does anyone know what this equation is?

Sorry for being so vague, but I am not sure how to look this one up myself. I've had a shirt with this on it for something like 20 years now and everyone I've shown it to says they don't know what it ...