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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0answers
37 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
4
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1answer
47 views

The completeness relation from QM in terms of inner products

I remember from QM that the completeness relation says $$ \sum_{n=1}^\infty |e_n\rangle \langle e_n | = I$$ so that $\langle x\mid y\rangle =\sum_{n=1}^\infty \langle x\mid e_n\rangle \langle e_n \...
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2answers
15 views

Which linear transformations are more abundant: dimension-increasing, preserving or decreasing?

My final aim is to understand the increase of Von Neumann Entropy in quantum systems by analyzing classes of unitary matrices in finite-dimensional Hilbert spaces. I'm following a potentially very ...
5
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1answer
98 views

Complementary text for mathematical Quantum Mechanics lectures

I'm looking for a text to complement Frederic Schuller's lectures on QM. His approach is very mathematical -- in fact it looks like the first 12 of 21 lectures are just about the mathematical ...
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0answers
7 views

Gradient of piece wise constant quantum control problem to steer system evolution to a target state

I'm looking for an exact gradient for the piece wise constant control of a quantum system to steer it towards a desired state at time T. It is worth mentioning, the Hamiltonians have been expanded ...
1
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1answer
26 views

Infinite propagation speed for the Schrödinger equation

I've seen many articles making reference to the property of the infinite propagation speed for the solution of the linear Schrödinger equation; but i can't find a book giving a 'good' definition or a ...
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0answers
22 views

Construct a master (possibly hypergeometric) formula from a family of formulas indexed by the half-integers and integers

I have a set of individual formulas ($a=1/2, 1, 3/2,\ldots,6$), each itself a function of an integer variable $k$, of increasing complexity. I would like to find a "master" formula (conjecturally of a ...
0
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1answer
18 views

Time (only) dependence with respect to the inner product of a wave function in $L^2(\mathbb{R})$

In my book "Quantum Theory for Mathematians" By B. Hall there is a discussion about the derivative of the inner product of a time-dependent wave functions $\psi(t)$ (note: no position dependence is ...
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0answers
13 views

Poisson bracket, Liouville operator puzzle

I am doing some baby Quantum mechanics (I have no formal training in the subject) and I keep getting the identity between two complex numbers $A$ and $B$ \begin{equation*} \overline{A}iB-A\overline{...
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0answers
22 views

Biorthogonality of vectors

This question is equal parts math and physics, though I chose to ask it here because I am more concerned with the mathematics behind it, rather than physical implications. Let $\hat{K}$ be a non-...
3
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1answer
78 views

What is the mathematical meaning of a quantum operator?

(Context: I am learning functional analysis using the book by Erwin Kreyszig "Introductory functional analysis with applications." The last chapter is dedicated to the applications of functional ...
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2answers
77 views

Quantum Mechanics: position and the separability of Hilbert space?

I would be pleased if someone could point out to me where I go wrong in the following sequence of statements: One model of quantum mechanics identifies states of a particle with normalized vectors (...
0
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1answer
33 views

Operators and Commutators in Quantum Mechanics

I'm trying to understand the terms "Operators" and "Commutators". Operators / Variables helps us to derive a differential equation that our wave equation must satisfy. Ex. Momentum Operator $P = \...
0
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0answers
33 views

Quantum theory linearly independent solutions

I'm trying to do the part of this qusetion where we need to find two linearly independent solutions to (2) of the given form. Is there a nicer way to do it other than just plugging it into (2). I was ...
0
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2answers
48 views

Quantum oscillator transition amplitude

I have a quick question regarding an equation in my textbook. It's about calculating the probability transition amplitude of a quantum oscillator. Why is this true? The difference between the first ...
0
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2answers
35 views

Non-Geometric Interpretation of the Dot or Inner Product.

I was wondering if there is a non-geometric interpretation of the dot product (or the inner product more generally). That is, an interpretation that has no concept of length and angle. My motivation ...
2
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0answers
18 views

integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want ...
0
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1answer
23 views

Comparing the definition of the Expectation Value to its application in QM

The definition of the expectation value for a continuous domain f(x) is given by $$<f(x)>=\int{f(x)p(x)dx}$$ where p(x) is the probability density function corresponding to {x}. In quantum ...
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0answers
30 views

Are there explicit formulas for the eigenvalues and eigenvectors of a generic 4x4 density matrix?

I have a 4x4 density matrix all of whose elements are nonzero. Its form is $$\begin{pmatrix} a & b & c & d \\ b^* & e & f & g \\ c^* & f^* & h & j \\ d^*...
0
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1answer
25 views

Solution of Schrodinger equation for free particle - How to eliminate mass variable

Probably missing something very obvious, sorry if this is a stupid question. I have to show the function $\psi(x, y) = \frac{1}{\sqrt{2\pi i t}} \int_{-\infty}^{\infty} e^{i(x-y)^2 / 4t} \psi_0 (y) ...
0
votes
1answer
31 views

Definite Gaussian/exponential integral

I've been revising some quantum mechanics, and I was wondering how I would calculate a specific standard deviation. The wave function I am working with is $\psi(x)=C(a,b)\exp(\frac{-(x-x_0)^2}{4a^2})...
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0answers
15 views

Transformation of inner product of wave functions under transformation of metric

Assume that we have a wave function $\psi(x)$ in the coordinate system $x$ in the Hilbert space $H_1$. The inner product of two states $\psi_1$ and $\psi_2$ are given as $\langle\psi_1|\...
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0answers
15 views

Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
0
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0answers
29 views

Normalize wave function and check if it is in Hilbert space

Consider a particle of mass m freely propagating within the box x ∈ [0, R]. Prepare the particle in the state corresponding to the wave function $ \psi (x) Asin(\frac{3\pi x}{2R}) cos(\frac{\...
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1answer
25 views

Quantum mechanics, operators acting on generic state

For a quantum mechanical harmonic oscillator of constant real mass m and frequency ω, define the following operators on the Hilbert space: $ h = a^{+}a $ $e = [\sqrt{-1 + a^{+}a}]a^{+} $ $ f ...
0
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1answer
24 views

Why is the Hermitian conjugate of the Fourier transform of an operator not the transform of the Hermitian conjugate?

It is defined that: \begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} \...
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2answers
49 views

Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
2
votes
1answer
22 views

Computate the commutator $[p^n,x]=-ihnp^{n-1}$

Computate the commutator of $[p^n,x]=-ihnp^{n-1}$. With $p=-ih \frac{\delta}{\delta x}$ the impulse operator. $h$ stands for $\frac{h}{2\pi}$. Answer: I do it with induction over $n$. For $n=1$ it ...
0
votes
1answer
20 views

If $L$ is a diagonalizable linear operator, why is $f(L)$ well-defined for $f: \mathbb{C} \rightarrow \mathbb{C}$?

In a book on quantum mechanics, I encountered a statement equivalent to the following. Suppose $V$ is a finite dimensional inner product space over $\mathbb{C}$. Let $L : V \rightarrow V$ be a normal ...
1
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1answer
35 views

Find the eigenvectors of a hermitian matrix as a function of angles

I have a problem which seems flawed to me. Consider the hermitian matrix M \begin{pmatrix} a/2 & b^* \\ b & -a/2 \end{pmatrix} with a>0 real and b complex. Let $\theta$,$\phi$ ...
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1answer
29 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
3
votes
1answer
46 views

Quantum Mechanics - Eigenvalue and Eigenvector of a Matrix

I'm attempting to find the eigenvalues and eigenvectors from the following matrix : \begin{pmatrix} -3\cos\theta&\sqrt{3}\sin\theta e^{iφ}&0&0\\\sqrt{3}\sin\theta e^{-iφ}&-\cos\theta&...
1
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1answer
30 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
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0answers
21 views

Intuitive interpretation of negative probabilities

I have heard that in quantum physics negative probabilities show up in certain distributions. Could you give an example that aids int he intuitional interpretation of a negative probability? For ...
2
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0answers
20 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
1
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1answer
36 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
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0answers
32 views

How to find the ground energy state solution in a quantum harmonic oscillator?

Recently, I came across a question which asks to solve the Schrödinger equation for a harmonic oscillator on $ [a, b] $ : $-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = ...
0
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0answers
33 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
0
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0answers
12 views

Separation of Variables PDE on Klein Gordon Equ

I have to use separation of variables on the 3-D Klein-Gordon equation: $ c^2 \bigtriangledown^2 \Psi - \frac{\partial ^2}{\partial t^2}\Psi + (\frac{mc^2}{\hbar})\Psi = 0 $ where $ \Psi (r,t) = \...
0
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2answers
56 views

N-Representation of an Operator

Calculating $\langle{n} \; {| \; \hat{X}^2 \; |} \; m\rangle$ in the N-representation, where $| \; m\rangle$ and $| \; n\rangle$ are harmonic oscillator states and $\hat{X} = \sqrt{\frac{\hbar}{2mw}}( ...
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0answers
11 views

Positive semi-definite vs. semi-positive definite?

I've heard and read the phrase positive semi-definite in many places. However, the only place I can recall seeing semi-positive definite is in my quantum mechanics text, John S. Townsend's $\...
2
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1answer
83 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
2
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1answer
40 views

A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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0answers
36 views

Dirac Notation: What Do $\langle u|v \rangle$ and $\langle u|T|v \rangle$ Represent?

I post this hoping for clarification, and particularly in a context of linear algebra without too much mention of matrices. (1) Let $V$ be a (perhaps infinite dimensional) Hilbert space and $V'$ its ...
3
votes
1answer
42 views

Singular value decomposition of sum of single particle operators

here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: \begin{equation} L = (L_0\otimes \mathbb{I} + ...
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0answers
35 views

How to prove the given property of spherical harmonics?

How to prove: $$ \int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0 $$ unless $l, l',$ and $l''$ are integers that can ...
5
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0answers
72 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + yR(\...
2
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0answers
23 views

Probability distribution obtained by repeatedly sampling $S_x,S_y$ on a spin-$S$ system

While trying to rework an upcoming quiz problem for a quantum physics course, I came up with the following question which turned out to be harder than I expected. (Note: I take $\hbar =1$ in all ...
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0answers
22 views

Unitary transformation with eigenvectors of fourier basis

I'm trying to understand this statement made in https://users.cs.duke.edu/~reif/courses/randlectures/UVnotes/lec18.pdf in the last paragraph: "Since U is multiplication by a group element, the ...
4
votes
2answers
33 views

Expectation of Complex Operators

Given an operator $\hat{\alpha}$, how do we obtain, $$ \sqrt{ \left\langle \left( \hat{\alpha} - \left\langle\hat{\alpha}\right\rangle \right)^2 \right\rangle } = \sqrt{ \left\...