# Tagged Questions

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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### Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output. [on hold]

I've been trying to solve the following Schrödinger equation numerically, -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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-...
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### 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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### To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
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### Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
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### 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 (...
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### Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
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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 $P = \... 2answers 48 views ### Quantum oscillator transition amplitude I have a quick question regarding an equation in my textbook. It's about calculating the probability transition amplitude of a quantum oscillator. Why is this true? The difference between the first ... 0answers 33 views ### Quantum theory linearly independent solutions I'm trying to do the part of this qusetion where we need to find two linearly independent solutions to (2) of the given form. Is there a nicer way to do it other than just plugging it into (2). I was ... 2answers 35 views ### Non-Geometric Interpretation of the Dot or Inner Product. I was wondering if there is a non-geometric interpretation of the dot product (or the inner product more generally). That is, an interpretation that has no concept of length and angle. My motivation ... 0answers 18 views ### integral of product of three basis functions and Clebsh-Gordan coefficients Suppose I have an orthonormal basis$\{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_2inner product). I want ... 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 ... 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^*... 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) ... 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})... 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|\... 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 ... 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} \... 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{\... 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 ... 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 that1=||AA|| \leq ||A||^2$$so that ||A|| \geq 1. But can anything else be ... 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 ... 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 ... 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 ... 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 ... 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&... 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 ... 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 ... 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 ... 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 ... 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 = ... 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}}( ... 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)$$we see that X has ... 0answers 12 views ### Separation of Variables PDE on Klein Gordon Equ I have to use separation of variables on the 3-D Klein-Gordon equation: c^2 \bigtriangledown^2 \Psi - \frac{\partial ^2}{\partial t^2}\Psi + (\frac{mc^2}{\hbar})\Psi = 0 where \Psi (r,t) = \... 1answer 48 views ### reduced density matrix for the given composite system Given the composite system of two qubits$$ |\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle)$with the density matrix of the composite ... 1answer 42 views ### Singular value decomposition of sum of single particle operators here is my question. Suppose to have an operator$L$in a composite Hilbert space$A\otimes B$which can be written as sum of single particle operator as: L = (L_0\otimes \mathbb{I} + ... 0answers 11 views ### Positive semi-definite vs. semi-positive definite? I've heard and read the phrase positive semi-definite in many places. However, the only place I can recall seeing semi-positive definite is in my quantum mechanics text, John S. Townsend's$\...
The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...