# 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.

929 questions
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### What geometry or topology best embodies the nonlocality of quantum entanglement?

I am a Princeton physics major. What geometry or topology best embodies the nonlocality of quantum entanglement? https://en.wikipedia.org/wiki/Quantum_entanglement: "Each particle cannot be ...
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### Derivative $\Lambda(\phi,\psi)=\frac{\langle\phi|A|\psi\rangle}{\langle\phi|\psi\rangle}$

I' m so sorry I found this expression on a handwritten sheet so I would like to check that it has sense and exactly what it means because I have not found a similar expression on any book. Let the ...
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### Schrodinger equation $iψ'(t) = H(t)ψ(t),\,\,\,\,\,\,\,\,\,\, ψ(t_0) = ψ_0,$

(Quantum Mechanics). A quantum mechanical system which can only attain finitely many states is described by a complex-valued vector $ψ(t) ∈ \mathbb{C}^n$. The square of the absolute values of the ...
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### Which Area of Math Contributes to Quantum Theory

Hello Math Community, Thank you for taking the time to read my question. It is much appreciated. I'm curious as to which branch of mathematics would help develop our understanding of quantum theory ...
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### Schrödinger Equation- ODE [closed]

Problem 3.29 from Gerald Teschl ODE. Let the Schrödinger Equation, $$i\psi'(t)=H(t)\psi(t),\ \psi(t_{0})=\psi_{0},$$ where $H(t)$, is a self-adjoint matrix, that is, $H(t)^{*}=H(t)$ . Show that the ...
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### Rotation and Translation Operators

Let $T_y(a)$ be a translation operator of a displacement $a$ parallel to the y-axis. In other words, $$T_y(a)\vec{r}=\vec{r}+a\;\hat{y}$$ If $R_x(\theta)$ is a rotation of $\theta$ around the x-...
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### $H_{SO}=f(r)\overrightarrow{S}\cdot\overrightarrow{L} \,\,\,$ $[H_{SO},L_z] \neq 0$

Let $$H_{SO}=f(r)\overrightarrow{S}\cdot\overrightarrow{L}$$ I know that: $$[H_{SO},L_z] \neq 0$$ $$[H_{SO},S_z]\neq 0$$ but I can not get or find the precise result.
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### Reduced density operators of a pure Bipartite state?

In (Bellucci, 2010; pg89) it is said (my wording): A pure bipartite state is separable if and only if the two reduced density matrices are pure. Proving the "only if" part is easy but I am ...
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### Number theory to estimate lower bound of spectrum in quantum mechanics?

I recently worked on the following idea: Eigenvalue of an Euler product type operator? Summary of the idea We represent numbers by infinite dimensional matrices such as $3$ will have all $0$s except ...
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### Closed form/ meaning of sum of geometric series of operator exponentials

I'm taking my first class on quantum mechanics right now and we've been using various operators throughout it; one operator we derived was the displacement operator, $$e^{a\frac{d}{dx}}f(x)=f(x+a)$$ I ...
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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. ...
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### Proof of identity using polynomial operator and commutators

Suppose x is a real number, and $\hat{A}$ and $\hat{B}$ are two non-commuting operators. A polynomial operator is defined by $$g(\hat{B}) = \sum_{n=0}^\infty a_n \hat{B}^n,$$ where $a_n$ are real ...
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### Perturbation of eigenvalues of Schrodinger Operator.

Suppose I have a sequence of Schrodinger opertors $$T_n=-\Delta +V_n$$ acting on (a subdomain of) $L^2(\mathbb{R}^d)$. Suppose that I view them as perturbations of the operator $$T=-\Delta+V$$ ...
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### Quantum Chemistry book recommendation. [closed]

I am trying to learn quantum chemistry. I have an extensive background in math and physics, so I'm looking for a book that makes full use of whatever physics and mathematics is relevant to this ...