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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Cayley's expansion

Is Cayley's expansion $$\exp(-iH\delta t) \psi(x,t)=\frac{1-\frac{i\delta t}{2}H}{1+\frac{i\delta t}{2}H}\psi(x,t)$$ valid for any operator $H$? What conditions should $H$ fulfill?
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States in light cone string theory

Currently I'm trying to understand string theory in the light cone quantization. I just have had a look into Polchinski (Vol. 1, Introduction to the bosonic string), because – as far as I could see – ...
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QM rotation operator

I have seen the proof that for fermions a rotation of $2 \pi$ does not return a spin angular momentum eigenstate to its original form, but instead multiplies the wavefunction by $-1$. Here is an ...
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Fock Space and fermionic annihilation & creation operators

I have been trying very hard to understand, I am reading Ballentine's book on this topic, but I need help: I realized that I don't understand how many particle states work with the creation & ...
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A conjecture about the Møller operator

Consider the Møller operator $$ \Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t } , $$ Now, suppose a state $\psi $ is located far away from the potential $V = H- H_0$. I feel that ...
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Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
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What is the correct way to treat operators that has “time” in QM? [duplicate]

I don't know if this question has already been resolved but considering that $i\hbar\partial_t$ is the energy operator, and $\partial^2_t$ is the waves operator (or helmholtz), I can't accept that $t$ ...
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What is the expectation value of the position times momentum operator?

Should I write the expectation of the position times momentum operator as: $$\langle xp\rangle = \langle \psi|x (-i\hbar \partial_x) |\psi \rangle$$ or $$\langle xp\rangle = \langle \psi| (-i\hbar ...
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What is the missing step in this result regarding the creation operators in Fock space?

In the above extract from Simons and Altman: Condensed Matter Field Theory, I am having trouble getting from (2.3) to (2.4) in the case of Fermions (ζ=-1 and the n(subscript i) values are modulo 2). ...
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Bounded operator - definition?

As mentioned also in Bounded and Unbounded Operator, an operator $A$ is said to be bounded, if $$\|Af\|\leq k \|f\|,$$ where the constant $k$ does not depend on the choice of $f$ (let us consider a ...
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Showing a measurement operator has a particular form

I came across an exercise (Ex 1.16) in 'Quantum Measurement and Control' by Wiseman and Milburn that I am having some trouble with. Suppose we have some system $S$ coupled with two meters in states ...
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Wave functions as $x$ goes to infinity

This problem emerged when I was going through some QM exercises: I've been asked to find the commutator $[A,B]$ where $A,B$ are defined as $$A\psi(x)=x\frac{\partial }{\partial x}\psi(x),$$ ...
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Reducing unitary evolution operator of a two-spin system to the evolution operator of one of the spins

Consider a system of two spins $s_1$ and $s_2$, each of which can be in one of two states, represented by 0 or 1. A basis for the Hilbert space of this system would be {|0,0>,|0,1>,|1,0> and |1,1>}, ...
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108 views

If I want to determine a particle's momentum or position, do I get this information from the wave function?

I am confused about how one measures the dynamical variables (eg position) of a particle. I thought the wave function $\Psi(x,t)$ was the probability amplitude and $|\Psi(x,t)|^2$ represents the ...
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179 views

How to prove that the position operator in momentum is $i\hbar \partial/\partial p$ - One Missing Sign [duplicate]

I am trying to prove that the position operator in momentum space is $i\hbar \partial/\partial p$ but my derivation is missing one sign. Can someone spot the error? Start with $$<\hat x> ...
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What am I REALLY doing when I take the Fourier transform of the momentum operator

I was playing around with some equations and found a surprising relationship when I took the fourier transform of the momentum operator Define $\hat P = \frac{\hbar}{i} \partial_x$, then $F(\hat P) = ...
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What is the difference between a parameter, a variable, and an operator in QM?

On the question why time isn't an operator, people will usually say that time is a parameter in QM (Time as a Hermitian operator in QM?) and not a variable. Can someone please distinguish between a ...
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Multivariable functions of Grassmann numbers

I'm trying to derive the closed form of the fermionic coherent state defined by the relation: $$ f_i|\vec{\eta}\rangle = \eta_i |\vec{\eta}\rangle \tag{4.10} $$ My book (Atland and Simons, Condensed ...
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Finding the matrix representation of a superoperator

I am trying to express superoperator (e.g. the Liouvillian) as matrices and am having a hard time finding a way to do this. For instance, given the Pauli matrix $\sigma_y$, how do I find the matrix ...
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Why isn't the Heisenberg uncertainty principle stated in terms of spacetime?

As I understand it, there are two "versions" of the Heisenberg uncertainty principle: Position-Momentum uncertainty \begin{equation} \sigma_x \sigma_p \geq \frac{\hbar}{2} \end{equation} where ...
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A series of bound states covering an interval

Generally, the bound states (normalizable eigenvectors) of a Hamiltonian have discrete eigenvalues. Is it possible for the eigenvalues to cover an interval? Say, $(a,b)$? That is, for each $E \in ...
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About the orthogonality of the Hamiltonian eigenstates for the the continuous energy spectrum

I would like first to describe a strange case that I encountered. $ \ \ - $ I solved the Schrodinger equation with a potential barrier (a potential well limited by a finite height wall which decrease ...
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Regarding calculations with plane waves

I'm dealing with some basic calculations with plane waves and I'm having some trouble with an idea. It has been said in another question that if you take to momenta like, for example ...
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Expectation value with plane waves [duplicate]

Hey guys Im a little confused with the concept of plane waves and how to perform an expectation value. Let me show you by an example. Suppose you have a wave function of the form ...
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Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?

I am reading Why we do quantum mechanics on Hilbert spaces by Armin Scrinzi. He says on page 13: What is new in quantum mechanics is non-commutativity. For handling this, the Hilbert space ...
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Multiplication properties about trace of two operators [closed]

Consider two operators $A$ and $B$, their functions $e^A$ and $e^B$ and a basis that mutual diagonalizes $A$ and $B$. Can I say that $$Tr\left[e^Ae^B\right]=Tr\left[e^A\right]Tr\left[e^B\right]~?$$ ...
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Tensor operators and transformation of $O^s_{\ell}|j,m,\alpha\rangle$

In H. Georgi's Lie Algebras in Particle Physics one defines a tensor operator transforming under the spin-$s$ representation of $SU(2)$ as the set of operators $O^s_{\ell}$ (for $\ell=-s...s$) such ...
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Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
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79 views

Different kinds of trace for statistical ensembles

In the chapter 7 of the book "A Modern Course in Statiscal Physics" by L. Reichl, we found $Tr[\hat{\rho}]=1$ for microcanonical ensembles and $Tr_N[\hat{\rho}]=1$ for canonical and grandcanonical ...
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Rectangular window $\psi$ wave-function and the calculus of $\langle p^2\rangle$ for it

I'm currently considering a rectangular window $\psi$ function: $$ \psi(x) = \begin{cases}\left(2a\right)^{-1/2}&\text{for } |x|<a \\ 0&\text{otherwise.} \end{cases} $$ I am interested in ...
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Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
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Calculation of $\langle p\rangle$ and $\langle p^2\rangle$ for wave function [closed]

Given the wave function $$\psi(x)=A\exp\left[-a \left(\frac{mx^{2}}{\hbar}+it\right)\right]$$ I would like to calculate $\sigma_{p}$. \begin{align}\langle p\rangle &=\int ...
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How does the Hermiticity of an operator imply that functions have an expansion in in multiple bases?

In Shankar QM it is stated that since the $\boldsymbol K$ operator is Hermitian, vectors, which are expanded in the $\boldsymbol X$ basis with components $f(x) = \langle x | f \rangle$, must have an ...
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Do we need an orthonormal basis in Quantum Mechanics?

I was wondering if it is important in Quantum Mechanics to deal with operators that have an orthonormal basis of eigenstates? Imagine that we would have an operator (finite-dimensional) acting on a ...
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Proving that conservation of momentum doesn't apply to electron in H-atom

To prove that the conservation of linear momentum doesn't apply to electron in H-atom, is it sufficient to show that angular momentum operator ($\hat L$) and momentum operator ($\hat p$) do not ...
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Can changing representation change the meaning of density operator?

I had posted a question What is the actual meaning of the density operator?. After that I understood that if I have the expression of a density operator $$\rho=\sum_{i=1}^{i=k}p_i|\psi_i\rangle ...
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Basis spin states

We are given a system of $N$ spin states and the following (non-hermitian) Hamiltonian $$H = \frac{N \hbar \nu}{2M} \sin(\alpha)+ \sum_{i=1}^N \frac{\hbar \omega_i }{2} \sigma_{z,i} + \frac{\hbar \nu ...
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Bopp operators and Wigner-Weyl representation

I am learning about the Wigner-Weyl transformations to move a $c$-number Lindblad operator $A(x,p)$ back into operator form. As far as I know, to move back and forth normally requires a four variable ...
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Can one representation of a projector operator be re-arranged to get another?

I have a vector space $V$ and a subspace of $V$, $W$. Let $P$ be the projection operator for subspace $W$. Also let the dimension of $W$ be $d$. Also I have two orthonormal basis $(a_1,a_2,...a_d)$ ...
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Group representation acting on operators (QFT)

I have found in many texts the following statement: Let $T_g$ be a representation of a group (of transformations, e.g. rotations, translations, Lorentz transformations ) acting on a given Hilbert ...
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Does quantum mechanics only deal with Hermitian/self-adjoint operators?

Whatever matrix I am seeing in quantum mechanics are all Hermitian matrices. We are using their eigenvalues for different types of work. Fortunately all their eigenvalues are real. Have you ever seen ...
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The Hermitian operator can have many outer product representations?

Let $H$ be a Hermitian operator, so it can be written as $$H=\lambda_1P_1+\lambda_2P_2........\lambda_kP_k,$$ where $\lambda_i$ are eigen values and $P_i$ corresponding projector operators for the ...
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Charge operator and the Goldstone boson

Can you explain a one question from Goldstone theorem about charge operator, what does it mean when theory said that charge operator annihilate vacuum and even it create new state of vacuum, which is ...
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What is the physical significance of the two integration constants that appear in the ladder operator decomposition of the Quantum Hamiltonian?

If I have a simple one dimensional Hamiltonian of the form \begin{align} H = V - \partial_x^2 \end{align} and if I know one zero energy state solution $H\psi_0=0$ then I can use the Wronskian to ...
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Is there a connection between Lie Groups and observable quantities in physics?

Good evening everybody. I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...
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Is there a simple way of finding the eigenstates of the creation and annihilation operator in QM?

How can I find the eigenstates of creation and annihilation operator in QM? My attempt: Such eigenstate will obey: $$ a^{\dagger} |\psi \rangle = \alpha |\psi \rangle. $$ We can expand $|\psi ...
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Finding a basis for minimal representation of a wavefunction (extracting symmetries)

I asked something like this on Math StackExchange, but now that I think about it, this probably belongs better over here. I want to find all linear operators (non necessarily hermitian) $\{\hat{A}\}$ ...
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Divergence in the total momentum opertator in QFT

The classical expression for the total momentum operator is $$P^{i} = -\int d^3x \, \pi(x) \, \partial_{i} \phi(x),$$ which, after second quantisation, using $$\hat{\phi}(x) = \int \frac{d^3k}{(2 ...
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QM: operators like $\hat{\mathbf{r}} \cdot \hat{{\mathbf{p}}}$

How would we treat an operator of the form $ \hat{\mathbf{A}} \propto \hat{\mathbf{r}} \cdot \hat{{\mathbf{p}}} $ ? Would it have eigenstates that are also eigenfunctions of position and/or momentum? ...
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Where does this commutator relation come from?

What is the origin of this relation: $$ [H,a_n^\dagger] = \epsilon_n a_n^\dagger $$ for Hamiltonian $H$, creation operator $ a_n^\dagger $, and eigenvalue $ \epsilon_n $. The square brackets denote ...