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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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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1answer
56 views

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 $$Tr\left[e^Ae^B\right]=Tr\left[e^A\right]Tr\left[e^B\right]~?$$ ...
0
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1answer
73 views

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 ...
1
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0answers
78 views

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 ...
2
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1answer
65 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 ...
4
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2answers
120 views

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 ...
4
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0answers
117 views

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 ...
1
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2answers
96 views

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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1answer
40 views

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 ...
2
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3answers
266 views

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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2answers
130 views

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 ...
0
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1answer
48 views

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 ...
1
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0answers
67 views

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 ...
0
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0answers
53 views

Hamiltonian of a linear chain of atoms and canonical quantization

When we want to re-formalize the Hamiltonian of a linear chain of atoms which has the following form: , we define the ladder operators as: and we use the following relations: to show that we ...
0
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0answers
54 views

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 ...
1
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2answers
54 views

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 $(a_1,a_2,...a_d)$ ...
2
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2answers
101 views

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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3answers
134 views

Does quantum mechanics only deal with Hermitian/self-adjoint operators?

Whatever matrix I am seeing in quantum mechanics all 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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3answers
140 views

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 ...
0
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1answer
48 views

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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37 views

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 ...
3
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1answer
128 views

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 ...
0
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0answers
129 views

Calculating the square of angular momentum operator in cylindrical coordinates

I want to evaluate the square of angular momentum, $L^2$, in cylindrical coordinates. I found components of $L$ in cylindrical coordinate. How can I find eigenvalue and eigenfunction of $L_z$?
2
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1answer
151 views

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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1answer
36 views

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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1answer
59 views

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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2answers
128 views

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? ...
0
votes
1answer
86 views

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 ...
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1answer
53 views

Wigner's Theorem and discrete Symmetries

According to my skript: A pure state is a ray: $\quad$ $\{λψ\}$, where $ψ ∈ \mathcal H$, $||ψ|| =1$ fixed and $λ ∈ \mathbb C$, $|λ| = 1$. Pure states are uniquely given by 1-dimensional orthogonal ...
0
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1answer
65 views

How can I write the anderson hamiltonian as a matrix? [closed]

How can I write this Hamiltonian: $$ H = \sum E_d \hat{n}_d + \sum_k \epsilon_k\hat{n}_k + \sum_k V_{kd} (\hat{a}^\dagger_k \hat{a}_d + \hat{a}^\dagger_d \hat{a}_k) $$ in matrix form using its ...
2
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0answers
95 views

Lorentz transformation - Bjorken & Drell

I'm trying to derive (14.25) in Bjorken & Drell (B&D) QFT. This is $$\tag{14.25}U(\epsilon)A^\mu(x)U^{-1}(\epsilon) = A^\mu(x') - \epsilon^{\mu\nu}A_\nu(x') + \frac{\partial ...
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2answers
290 views

How did the operators come about?

This relates a little bit to my previous question (Experimentally, what categorizes a measurement as corresponding to a certain observable?), but it's different in a way and more historical. One of ...
4
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198 views

How does Dirac define the representative of $\{\langle\phi\frac{d}{dq}\}\psi\rangle = \langle\phi\{\frac{d}{dq}\psi\rangle\}$

On pate 89 of Dirac's book, The Principles of Quantum Mechanics, he writes: Let us treat the linear operator $\frac{d}{dq}$ according to the general theory of linear operators of section 7. We ...
6
votes
2answers
192 views

Does the Hermitian operator $H=-\frac{d^2}{dx^2}$ have imaginary eigenvalues?

In quantum mechanics, Hermitian operators play a very important role because they possess real eigenvalues. Considering $-\frac{d^2}{dx^2}$, it is a Hermitian operator (Actually it's the simplest ...
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2answers
150 views

Why is quantum mechancis is not content with symmetric operators, but wants self-adjoint operators?

A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal. These are exactly what we want for a physical observable. I think ...
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1answer
227 views

Discrete vs Continuous spectra of operators [duplicate]

Why is it that if an operator $Q$ has a discrete spectra, that the eigenfunctions are all in Hilbert space? Why is it that if the spectrum is continuous we automatically know that the eigenfunctions ...
4
votes
4answers
269 views

What is the right order of creation operators?

I started to learn some basics of second quantisation and specifically its use in quantum chemistry. Currently I'm reading this book by Péter R. Surján, and here is small excerpt from it. If one ...
1
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2answers
356 views

Hilbert space and Hamiltonians

Assume a system described by a Hamiltonian H, and assume that the eigenstates of H, $φ_i$(r) are integrable in absolute square. We say that these states belong to a Hilbert space (they can even form a ...
0
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1answer
61 views

Hamiltonian acting on sum operator

I am following a derivation in a book. It is implementing a state $|{\psi}\rangle$ into the eigenvalue equation $\hat{H}|{\psi}\rangle=E|{\psi}\rangle$. The $|{\psi}\rangle$ term contains a ...
0
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1answer
163 views

Spin operators commutation

Why do the spin operators $ S_{x1}$ and $S_{x2}$ of two particles along the $x$-axis commute i.e $S_{1x}S_{x2}-S_{2x}S_{1x}=0 $ ?
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1answer
47 views

Physical observables and hermiticity

Is it necessary for an operator to be Hermitian in order to be a physical observable or is it just sufficient that the operator obeys the eigenvalue equation? If I were to check whether an operator is ...
0
votes
3answers
213 views

About the postulates of quantum mechanics and self-adjointness

I am a freshman trying to understand the very basics of quantum mechanics but I met barriers at the beginning. What really matters is the postulates of quantum mechanics and their relationship with ...
0
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1answer
168 views

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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1answer
117 views

Projection operators in a direct product space

The things I'm pretty sure I understand: Let's say I have a single particle hamiltonian $H$ represented by a $2$x$2$ matrix, so it has two eigenstates $|\lambda_1\rangle$ and $|\lambda_2\rangle$. I ...
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1answer
67 views

What is the relation between renormalization and self-adjoint extension?

What is the relation between renormalization and self-adjoint extension? It seems that a renormalization scheme can be rigorously treated mathematically using the self-adjoint extension theory. Is ...
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1answer
31 views

Addition of angular momenta, coefficient in the |10> state

In Griffith's text, they apply the lowering operator on the |$11\rangle$ state to get the |$10\rangle$ state. They show this result in two forms on pg. 185: $S_{-}\left(\uparrow\uparrow\right) = ...
2
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2answers
43 views

String Excited States / Spectrum

I am working through David Tong's lecture notes on string theory. The bit I am stuck on is page 41/42 in Chapter 2. The PDF file is here. In 2.3.2 "The First Excited States" on page 41/42, he says: ...
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1answer
73 views

Eigenvalues of Angular Momentum in Quantum Mechanics

The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...
0
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2answers
233 views

Creation and annihilation operators in Hamiltonian

If I find a Hamiltonian $H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_k V_k a_k^{\dagger} a_k$ then I was wondering: As far as I know this is many body theory and so these operators act on ...
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50 views

Finding the expectation value of the annihilation operator with respect to a given state

Using dirac notation we were given a state vector $$|\Psi(t=0)\rangle = A\sum\limits_{q=0}^Q \frac{1}{(q+i)} |\phi_q\rangle$$ Where $\phi$ is part of a complete orthonormal set. I found the ...