# Tagged Questions

In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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### Both Eigenvalues and Operators are “Observables”? [duplicate]

I am having a bit of difficulty wading through the what seems to be multiple usages for Observables in Quantum Mechanics. " Mathematically observables are postulated to be Hermitian operators.. " ...
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### Why isn't the time-derivative considered an operator in quantum mechanics? [duplicate]

Based on my understanding when doing quantum mechanics we deal with a small set of mathematical objects: namely scalars, kets, bras, and operators. But then in the Schrodinger equation we have this ...
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### Find the expression of $j_+$ and $j_-$operators [closed]

I have just started to do exercises about quantum mechanics and I have some difficulties. I have a particle with spin $j=1$ and the Hamiltonian is $H=\gamma(j_xj_y+j_y j_x)$ (where $\gamma$ is a ...
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### Can the inverse operator be expressed as a series?

I've seen the claim that a function of an operator can be defined as a series. For example, say $A: H_1 \mapsto H_2$ is an operator. Then $$e^A \equiv \sum_{n=0}^\infty \frac{A^n}{n!}. \tag{1}$$ In ...
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### Do I have some freedom when I define the quantum SHO ladder operators? [closed]

I tried to solve the quantum harmonic oscillator via the operator method. After doing it and looking up the solution I noticed that for some reason the ladder operators got an additional factor of (i) ...
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### Continuous spectrum of hydrogen atom

I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.
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### A particle in a 1D box: what is the meaning of velocity?

In the box $x = 0$ to $x = L$, $V = 0$, and for $x < 0$ and $x > L$, $V = \infty$ (infinite potential well). The eigenvalues of the Hamiltonian are: $$E_n = \frac{n^2 h^2}{8L^2} \, .$$ Since ...
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### Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
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### How to write electron hole Hamiltonian into quasi-boson form?

V Chernyak, Wei Min Zhang, S Mukamel, J Chem Phys Vol. 109, 9587 (can be freely downloaded here http://mukamel.ps.uci.edu/publications/pdfs/347.pdf ) Eq.(2.2), Eq. (B1) Eq.(B4)-(B6). When I substitue ...
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### Smoothness of the energy levels of a generic Hamiltonian

Let us take an Hamiltonian $H(\xi)$ which depends on a set of parameters $\xi$, and assume that the matrix elements $h_{ij}(\xi)$ of the Hamiltonian are smooth complex functions of the parameters $\xi$...
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### Angular Momentum Operators - Commutation Relations

I was going over past PGRE exam questions, and came across this one. The components for the angular momentum operator $\mathbf{L}=(L_x,L_y,L_z)$ satisfy the following commutation relations. \...
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### The $n$-th root of the NOT gate

I simply can not find material containing facts about the $n$-th root of the NOT gate and it's realization in Q.M. and also in C.M.. Does anyone have material? A comparison of the $n$-th root NOT ...
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### Eigenstates of sum of creation and annihilation operators

Does the operator $a+a^\dagger$ have eigenstates? If yes, what are they?
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### Mode operators in the Virasoro algebra

This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the ...
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### Probability flux

I was reading a text on Quantum Mechanics in which it said that $$\int{d^3 x \, j(x,t)} = \frac{\langle p\rangle}{m},$$ where $\langle p\rangle$ is the expectation value of the momentum operator at ...
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### Quantum Mechanics - Lowering Operator [closed]

Let $a$ be a lowering operator. Show that $a$ is a derivative respects to raising operator, $a^\dagger$, $$a = \frac{\textrm{d}}{\textrm{d}a^\dagger}$$ Can someone please explain how to prove the ...
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### Time reversal symmetry and real symmetric Hamiltonian matrix

In the literature (like those in quantum chaos), it seems that time-reversal symmetry implies that the Hamiltonian of the system is a real symmetric one, instead of just being complex Hermitian. Is ...
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### Separability of the Hilbert space: countable orthonormal basis vs. continuous spectrum

Hilbert spaces are mostly assumed to be separable. A Hilbert space is separable if and only if it admits a countable orthonormal basis. How does this fit together with the possible existence of the ...
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### Operators is a infinite dimensional matrix, how can it multiply by a wave function that is a n*1 (n is finite) matrix

My confusion started from thinking the quantum superposition principle. Several website say that the quantum superposition means all state can be represented as infinity superposition of orthogonal ...
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### Wick's Theorem: Why is the vacuum expectation value of uncontracted operators zero?

I'm am right now reading Chapter 4.3 (Wick's Theorem) in Peskin & Schroeder. It is said that In the vacuum expectation value, any term in which there remain uncontracted operators gives zero (...
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### “Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit ...
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### What is the condition for local operations on bipartite entangled state?

I have an entangled state between Alice and Bob $|\psi\rangle_{AB}$ ( both Alice and Bob have states in Hiblert space of dimension $n$ ). Alice and Bob can only perform local meaurements. I assumed ...
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### Commutators and Operators [closed]

Is commutator of two operators an operator? I searched google but still got no success! I'm very curious to know the answer to this!
In the Principles of Quantum Mechanics, Dirac states that all linear operators $\alpha$ over our vector field (over the complex numbers) can be expressed as the sum of a real and an imaginary part \$\...