# 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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### log det on density matrix plus identity

A very naive question: given a pure quantum state $|\phi\rangle$, and the associated density matrix $\rho=|\phi\rangle\langle\phi|$, does there exist an efficient quantum operator/procedure that gives ...
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### Path Integral in QM - Fourier Transform with respect to a Function?

Consider the Fourier transform of a multivariable probability density function $Pr(\{x_n\})$, i.e. its characteristic function: $\int Pr(\{x_n\})e^{-i2\pi\sum\limits_{n}f_nx_n}\prod\limits_{n}{dx_n}$, ...
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### False Negative Probability of a probabilistic black box function

Suppose we have black-box access to a function that returns the roots of some unknown 5-degree polynomial in each invocation. (So there will be 5 roots.) But we also know that the function is ...
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### How to perform a functional integral?

I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
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### Hermitian and Observables

Let $H$ be an finite dim Hilbert space with $|\phi \rangle$ in H, $\langle\phi \mid \phi\rangle=1$, and $\rho \triangleq|\phi\rangle\langle\phi| .$ Let $A$ be an observable which is represented by the ...
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### Definition of the pullback of $L=Im(dS)$ and related questions on defintions

Def: We call a phase function $S: \mathbb{R}^n \rightarrow \mathbb{R}$ admissible if it satisfies the Hamilton-Jacobi equation. The image $L=Im(dS)$ of the differential of an admissible phase function ...
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### Is there a connection between graph theory and many worlds QM

Obligatory disclaimer that this is my first post and first contact with the community so please redirect me to better channels if these types of questions are not for this forum. So I was just naively ...
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### Identity regarding Pseudodifferential operator

In the paper I am reading, the following identity appears: $$e^{-it \Delta} f(x) e^{it \Delta} = f(x- 2it \nabla)$$ where $f \in \mathcal{D}(\mathbb{R}^{d})$ and $f(x)$ on the left hand side denotes ...
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### Why does the order of eigenvectors matter when changing a matrix to a different basis?

I'm solving a basic problem in quantum mechanics to change the matrix A into a representation in the basis consisting of eigenvectors of matrix B. And I've noticed that the new mateix A looks ...
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### What can I infer from this scalar product identity?

Suppose I have shown for a single vector $\lvert 0 \rangle$ that $$\langle 0 \rvert U^\dagger \phi U \lvert 0 \rangle=\langle 0 \rvert \phi \lvert 0 \rangle$$ where $\phi$ is a certain operator and $U$...
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### Matrices that are simultaneously Hermitian and unitary [duplicate]

My quantum mechanics professor was discussing the properties of Pauli matrices, their being both Hermitian and unitary. Then he made a remark that it is not possible to find three $n \times n$ ...
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### Can the solution of a real differential equation always chosen to be real?

I read somewhere, that the solution to a real differential equation can always be chosen to be a real-valued function. Is this true? In particular, I am interested in the stationary Schrödinger ...
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### Derivative of the action functional

I am self studying the book An Brief Introduction to Physics for Mathematicians. At the second page, the equation (1.6) says that the derivative of the action functional can be derived as  S'(x)(h) =...
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### Spherical harmonics as orthonormal basis in quantum mechanics

In this article https://mrtrix.readthedocs.io/en/dev/concepts/spherical_harmonics.html the following statement is given: Spherical harmonics are special functions defined on the surface of a sphere. ...
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### Calculating Error for using an approximation for the Schrodinger Equation Evolution Operator

I've devising a Crank-Nicolson scheme for the Schrodinger equation using a time-independent potential, $V(x)$. In devising this scheme the following equation is approximated and THEN discretized, ...
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### What is the relation between the classical and the quantum mechanical approach in calculating the Value at Risk?

What is the relation between the quantities in classical Value at Risk (VaR) calculation and the calculation via quantum computers? To specify this question, I would like to briefly explain my ...
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### Integrating an ODE with matrices

I've been watching David Deutsch's Lectures on Quantum Computation here https://www.youtube.com/playlist?list=PLqdVnC7OWuEcfKRZXsrooK_EPzwmWSi-N In Lecture 2, he gives these equations to show how X,Y ...