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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Time dependence of the displacement operator

I am following the derivation of the master equation (and application of this) in these lecture notes. Unfortunately I do not follow the step of eliminating the driving terms of the harmonic ...
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What is the algebraic form of the momentum eigenstate?

I'm asking this in the context of trying to verify the equation $a^{\dagger}_{p} \vert 0 \rangle = \frac{1}{\sqrt{2\omega_p}} \vert p \rangle$. So far I have calculated $\vert 0 \rangle = ...
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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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Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...
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Calculation of OPE in Polchinski

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
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Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
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Operator product expansion energy momentum tensor

We have the following equation from Polchinski (2.4.6) $$ T(z)X^{\mu}(0) \sim \frac{1}{z}\partial X^{\mu}(0) , \tag{2.4.6} $$ where $T(z)$ is defined as $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} ...
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Position operator for wave function of entangled system? [on hold]

When I read about EPR paradox, I have the following confusion. Consider the whole system, it is a pure state, so it can be described by a wave function. (For sub-system(one of the electrons), it is ...
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Quantum Mechanics - Induction Methods [closed]

Let $a$ be a lowering operator and $a^\dagger$ be a raising operator. Prove that $$a(a^\dagger)^n = n (a^\dagger)^{(n-1)} \, .$$ My professor suggested to use induction method with formula: $$\left( ...
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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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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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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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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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Different operator algebra for different systems [closed]

Up until now, I was under the impression that all wave functions lived in some Hilbert Space, perhaps due to the presentation of the material in my introductory quantum mechanics class. However, every ...
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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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How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
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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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“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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The Eigenstate Existence Problem in Dirac's book 'Principles of Quantum Mechanics'

In Chapter II of Dirac's book Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/eigenvectors ...
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Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
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Equation 2.27 from Pachos's introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
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Quantum mechanic particle

In non relativistic quantum mechanic, we are dealing with a problem involving a particle in one dimensional space, and it has been given the potential and it reads: ...
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Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
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Apparent spacetime dependence of creation and annihilation operators

I'm currently going through An Introduction to Quantum Field Theory by Hartmut Wittig I've stumbled upon. Having trouble with equation (2.29), I'm asking the question: Do creation and annihilation ...
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Constructing differential equation from arbitrary Hamiltonian

Suppose I begin with the time-independent Schrodinger equation $$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$ ordinarily we specify the function $V$ and then solve for a ...
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Ladder Operator killing state

So in my last question, @joshphysics showed me how to prove $K_\pm$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not ...
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Proof for Negele and Orland equation (2.34)

The equation (2.34) of Negele and Orland has $$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) = \frac{1}{2m}\left(\hat {\mathbf p} - \frac e c \mathbf A(\hat{\mathbf x})\right)^2.\tag{2.34a}$$ ...
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Why is only one quantity of angular momentum i.e. $L_z$ quantized & not $L_x$ & $L_y$?

This is quoted from Arthur Beiser's Concepts of Modern Physics: Why is only one quantity of $\mathbf{L}$ quantized? The answer is related to the fact that $\mathbf{L}$ can never point in any ...
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Where can I find a detailed derivation of the form of two body operators in the second quantization?

I've been looking around online for a couple hours now and I can't find a very informative derivation of the form for two body operators in the second quantization. Is there a resource online ...
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How to interpret vector operators in quantum mechanics?

To the point: How should I think about the equation $$\hat{\mathbf{x}}\mid\mathbf{x'}\rangle = \mathbf{x'}\mid\mathbf{x'}\rangle~?$$ Is it a triple of equations $\hat{x}\mid x'\rangle = x'\mid ...
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Do we get the same answer at any time if we measure a system's energy?

Schrödinger's equation says that the only allowed energy states of a system are the eigenvalues of the energy operator $H$. This means that if we measure the energy of the system at any time we ...
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Really how can an observable quantity be equal to an operator?

A wave-function can be written as $$\Psi = Ae^{-i(Et - px)/\hbar}$$ where $E$ & $p$ are the energy & momentum of the particle. Now, differentiating $\Psi$ w.r.t. $x$ and $t$ respectively, ...
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Particle in a box - speed probability distribution

Consider a particle in a box with infinite barriers. By solving the Schrödinger we can find the probability of finding the particle at some points in the box. How can we find the probability of ...
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Uncertainty relation for non-simultaneous observation

Heisenberg's uncertainty relation in the Robertson-Schroedinger formulation is written as, $$\sigma_A^2 \sigma_B^2 \geq |\frac{1}{2} \langle\{\hat A, \hat B\}\rangle -\langle \hat A\rangle\langle ...
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What is the effect of squeezing on the Husimi phase space representation or Q-function?

The effect of the squeezing operator \begin{equation} S = e^{- r (a^2 + a^{\dagger 2}) / 2} \end{equation} on a Wigner phase space representation or W-function of a system with density matrix $\rho$ ...
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Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
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Schaum explication of eigenvectors of Lz

In Schaum's Quantum Mechanics, in Chapter 6 Angular Momentum, they say "the eigenvectors of $L^2$ and $L_z$ are functions that depend on the angles $\theta$ and $\phi$ only; hence, we can represent ...
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How does an operator transform under time reversal?

We know that a time-reversal operator $T$ can be represented as $$T=UK$$ where $U$ is some unitary operator and $K$ is the complex conjugation operator. Then under time-reversal operation, a quantum ...
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Change of variables for integral operator

One can write the operator $L=(\sqrt{1-i\partial_x^2}-1)$, as an integral, that is $$(\sqrt{1-i\partial_x^2}-1)B(x,t)=\frac{i}{4\pi^2} \int_{-\infty}^{\infty}(\omega(k_o+\kappa)-\omega(k_o))e^{i ...
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What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that \begin{equation} [\hat{H}_1,\hat{H}_2] = 0, \end{equation} where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
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46 views

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!
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How to transform the Laplacian from momentum space to coordinate space

I'm working through some quantum mechanics problems with solution sets (attempting the problems then looking at the solutions to compare), and a little part of a solution has stumped me. I'm not sure ...
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On linear operators and their complex qualities

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 ...
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How are anti-unitary operators applied?

I was reading about anti-unitary operators from Wikipedia. They give an example of an anti-unitary operator: were $K$ is complex conjugate operation. $\sigma_y$ is defined with respect to two ...
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Commuting observables and CSCO's

I've been looking at some basic quantum mechanics all day in an attempt to better my understanding of the subject. While going over the proof that commuting operators are compatible, I started getting ...
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$\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute… or do they?

So $\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute: $ [ \hat{L_{x}}, \hat{L_{y}}] = i\hbar \hat{L_{z}}$ But, what if we perform this operation on a state such that: $\hat{L_{z}} \phi_{l, m_{l}} = ...
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Would $[\hat{Q},\hat{H}]$ correspond to an observable? [closed]

Would $[\hat{Q},\hat{H}]$ correspond to an observable? Where $\hat{Q}$ is an observable and $\hat{H}$ is the Hamiltonian. Surely that would just mean that $[\hat{Q},\hat{H}]$ would commute i.e. = 0?: ...
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Relationship between those two “exponentials”

Let $G$ be a Lie group and $L(G)$ it's Lie algebra. We know that every left-invariant vector field $X$ in $G$ is complete, and so one can consider the integral curve defined for all $t\in \mathbb{R}$ ...