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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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Quantum Hermiticity Bra-Ket notation please

If $A$ and $B$ are Hermitian operators, show that $$C~:=~i[A,B]$$ is Hermitian too. My work: $$\begin{gather} C=i(AB-BA) \\ \langle\psi\rvert C\lvert\phi\rangle = i\langle\psi\rvert AB\lvert\phi\rangle-i\langle\psi\rvert…
Anon
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Schrödinger equation on a torus

Morrey's Identity: Define the Torus: $\pi^{d}=\frac{\mathbb{R}^{d}}{2 \pi \mathbb{Z}^{d}}$ Let $m \in \mathbb{N} $ and $s\in \mathbb{R}$ such that $s > m + \frac{d}{2}$ . Prove that there exists $ c > 0$ such that for all $u \in H^{s}(\pi ^{d})$…
Fareed Abi Farraj
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What is the intuition behind quantization?

It has long irked me that quantum mechanics appears to be so bound up in mysticism, if that isn't too strong a word. It shouldn't be like that: it should be possible to follow the reasoning all the way back to some intuitively obvious axioms -…
j4nd3r53n
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How can we show that a map is a completely positive map?

I am doing a homework problem where I have to find whether the map $$ \rho ~\rightarrow~ {\rm tr}(\rho) I - \rho $$ is completely positive. If the map is not completely positive, a counter-example will work. My question is: What is the general…
Kishor Bharti
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Solving Schrodinger for harmonic oscillator(griffiths analytic method)

I was just getting into quantum mechanics. But I'm having a bit of trouble following Griffiths for the analytic method. It goes like so: The Schrodinger equation is: $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = E…
Dustin.
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Dirac Delta function identities

I'm using the definition of the one dimensional dirac-delta function, $\delta(x)$, being, $$\int_{-\infty}^{+\infty}\delta(x)f(x)dx = f(0) \tag1$$ and I'm doing a question which asks me to (via ordinary integral manipulations) show that, for $a \in…
Premez
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Calculating the operator adjoint to the partial time derivative

I want to show that for any $\mathcal{L}^2(\mathbb{C}^n\times\mathbb{R},\mathbb{C})$-function $\Psi(x,t)$ the differential operator $\partial_t$ is self-adjoint, i.e. $\langle\Phi|\partial_t\Psi\rangle=\langle\partial_t\Phi|\Psi\rangle$ for any…
Thomas Wening
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Proving an operator is hermitian

I am supposed to prove an operator G is hermitian but I seem to be proving it isn't. G is defined as $$G = -i(O-O^t)$$ where $O$ is some linear operator. If I take the transpose of the whole thing I get $$G^t = -i(O^t - O),$$ which is $-G$. Am I…
boudreaux
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Probability difference scattering potentials

Let $V_1(x)$ and $V_2(x)$ be two real potential functions of one space dimension, and let $m$ be a positive constant. Suppose $V_1(x)\le V_2(x) \le 0$ for all $x$ and that $V_1(x) = V_2(x) = 0$ for all $x$ such that $|x| > a.$ Consider an incoming…
MathematicianP
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2 Dimensional quantum particle

A quantum particle moves in 2 dimensions with Hamiltonian H: $ H = \frac1{2m} ((P_1 + \frac12 eBX_2)^2 + (P_2 - \frac12 eBX_1)^2) $ For constants $e,B,m$ with $e$ and $B$ nonzero. Show that the energy levels are of the form $ (n + \frac12)\bar h…
Fahrenheit997
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Question about Dirac notation

So from what I understand $\langle w | v \rangle=\vec w^* \cdot \vec v$. Ok. I'm fine with that notation. But then I've seen $\langle x | y \rangle=\delta(x-y)$ and $\langle x | p \rangle=e^{-ixp/\hbar}$. I can see that these are the…
got it--thanks
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Bell state problem

Let's consider a system composed of the tensor product of two qubits in the following states: $$|\Psi\rangle = \alpha |00\rangle - \beta |11\rangle$$ $$\rho_{AB} = \rho_1|\Psi^{-}\rangle \langle \Psi^{-}| + \rho_2|\Psi^{+}\rangle \langle \Psi^{+}| +…
Yau
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Quantum algorithm

Let's suppose that we have a quantum superposition state, and for a basis vector $|\Psi\rangle\in \mathcal{H}$, $$|\psi\rangle = \frac{\sqrt{N-1}}{\sqrt{N}}|\Psi_c\rangle + \frac{1}{\sqrt{N}}|\Psi\rangle$$ Where $|\Psi_c\rangle$ is orthogonal…
Yau
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Clarification on Energy Levels for a Hyperbolic Wave Function in Introductory Quantum Theory

I am working through an introductory problem that is part of an undergraduate course for Quantum Theory. I am self-studying currently. $\DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\tanh}{tanh}$ A particle of mass $m$ moving on the…
jcneek
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What is the Hilbert space for the Schrodinger equation for the hydrogen atom

Brain Hall: Quantum Theory for Mathematicians p.87 "One peculiarity of the physics literature on quantum mechanics is a conspicuous failure of most articles to state what the Hilbert space is." As a chemist I grew up solving the Schrodinger…
luysii
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