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Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.

Quantum mechanics, also known as quantum physics or quantum theory, deals with classical physical phenomena at quantum scales.

The precise nature of the subject has changed over the years. This article explains its current formulation.

Quantum mechanics aims toprovide a mathematical description of the dual particle-like and wave-like behaviors and interactions of energy and matter.

1699 questions
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Showing that $UU^*=D$ where $D$ is a diagonal matrix

I've been receiving downvotes and would like to know why? pretty much the title. I think this condition isn't generally true but maybe I'm missing something (my maths could be horribly wrong). I was just wondering if someone wiser had any insights…
Mugenbi
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Expecation for tensor products

We are told that $$Z|0 \rangle = | 0\rangle \\ Z | 1\rangle =-|1\rangle \\ X|0\rangle =|1\rangle \\ X|1\rangle =|0\rangle$$ and we have the state $$|\psi\rangle =|0\rangle |1\rangle +|1\rangle |0\rangle$$ The question is: Without doing an explicit…
Traceback
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Weyl transform of a product of operator

I am struggling with a demonstration about Weyl transform properties. In particular, I want to prove $$\tilde{(\hat{A}\hat{B})}(x, p) = \tilde{A}(x,p)\exp(i\Lambda/2\hbar)\tilde{B}(x,p)\,,$$ where $\Lambda$ is the Poisson bracket operator. I will…
Jonny_92
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Prove that $(\hat A \hat B + \hat B \hat A )$ is self-adjoint if $\hat A$ and $\hat B$ are self-adjoint

I'm not clear why, in case $\hat A$ and $\hat B$ are self-adjoint, then $(\hat A \hat B + \hat B \hat A )$ is also self-adjoint. In order to show that $(\hat A \hat B + \hat B \hat A )=(\hat A \hat B + \hat B \hat A )^\dagger$, I have seen that the…
Invenietis
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How to show that $[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$ for any operator $\textbf{v}$ constructed from $\textbf{x}$ and/or $\nabla$?

In Weinberg's Lectures on Quantum Mechanics (pg 31), he said that the commutator relation $$[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$$ is true for any vector operator $\textbf{v}$ constructed from $\textbf{x}$ and/or $\nabla$, where $\textbf{L}$ is…
TaeNyFan
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Find solution of this equation

$$u_{t}+u u_{x}=0$$ $$u(x,0)= \begin{cases} 1 & \text{if}\quad x<0 \ 0 & \text{if}\quad 0
User124356
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Change in potential of quantum harmonic oscillator

We know that for the 1D time-independent SE with potential $V(x)=\frac{1}{2} m \omega^2 x^2$, the solutions have energies $E_n = (n + \frac{1}{2}) \hbar \omega$. I've been attempting the following question: Suppose a particle of charge $q$ and mass…
T.H
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What is the difference between superposition and a function that randomly returns a value?

I am struggling to understand a few things about quantum physics. One of those things is superposition. If a photon (for example) is in superposition, there is an equal probability that it will exist in one state or another. I can write a computer…
Tom V
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Commutator of $p^2$ and $x$

I need to calculate $[p^2,x]$, with $p=-i\hbar\frac{d}{dx}$ This is what I've done: $$[p,x]=-i\hbar$$ $$[p^2,x]=[p,x]p + p[p,x]$$ $$[p^2,x]=-i\hbar p - pi\hbar$$ Now here's my question: I've seen solutions to this problem doing $pi\hbar=i\hbar p$,…
its
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Susskind exercise on quantum mechanics

On Susskind book, exercise 3.1 say to prove the following: "If a vector space is $N$-dimensional, an orthonormal basis of $N$ vectors can be constructed from the eigenvectors of a Hermitian operator." Susskind wrote that the proof is easy. From…
franchino
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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} \end{equation*} It is tnteresting to calculate the…
alex
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Eigenvalues of an operator

I am stuck on this: Deduce the eigenvalues of the operator $A^{\dagger} A$ are positive, where $$A^{\dagger} = -\frac{\text{d}}{\text{d}x} + \tanh(x)$$ $$A = \frac{\text{d}}{\text{d}x} + \tanh(x)$$ The point is that those are not matrices, in the…
Enrico M.
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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. I need to find the case where: …
MRT
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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 is. I guess I'm not surprised. Does anyone here…
Kai Qing
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Angular momentum operator, rotation in 3 dimensions.

The operator for the total angular momentum and its z-component can be written as $\hat{L}^2=-\hbar^2\hat{\Lambda}^2$ (where $\hat{\Lambda}^2=\frac{1}{sin^2\theta}\frac{\partial ^2}{\partial…
ChemGuy
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