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 ...

learn more… | top users | synonyms

2
votes
1answer
35 views

Product on Tensor Products

I'm trying to understand how products on tensor products work. For instance, in quantum mechanics, you have ($x$ tensor $y$) times ($z$ tensor $a$), where $x$, $y$, $z$, $a$ are all operators acting ...
2
votes
2answers
100 views

Why we cannot describe operator for force $F$ in quantum mechanics?

In quantum mechanics we describe operators corresponding to momentum but we don't define operator for force what is the reason behind it?
0
votes
1answer
88 views

How much information does the Hamiltonian contain in quantum mechanics? [closed]

Given a Hamiltonian, let's say of a many-body system, through the Schrodinger equation,in principle we can find the eigenfunctions and their corresponding eigenvalues (spectrum). Now given an ...
10
votes
2answers
586 views

How are anyons possible?

If $|ψ\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator $P$ which switches the states of the two particles. Since the two particles are ...
0
votes
0answers
58 views

How to find creation and annihilation operators? [duplicate]

I get confused when trying to find this. Please describe it as simply as possible, but keep in mind I have no budget whatsoever to pay for textbooks, so here goes: How do you find the creation and ...
0
votes
2answers
64 views

Why are the charge operator $Q$ and the baryon number operator $B$ unbounded?

A friend recommended me to read PCT, Spin and Statistics, and All That written by R. F. Streater and A. S. Wightman. In page 5 to 6, here's what the authors of this book have to say: [...] In ...
-1
votes
1answer
39 views
0
votes
2answers
43 views

Show that a function takes the following form using the definition for the function of an operator

If $f(z)$ is a function with a Taylor series expansion $$f(z)=\sum _{ n=0 }^{ \infty }{c_n z^n },$$ then we define $$f(M)=\sum _{ n=0 }^{ \infty }{c_n M^n }.$$ First consider ...
0
votes
2answers
71 views

The delta function as an eigenfunction of the position operator explanation

$\delta (\textbf{r})$ can be interpreted as a wavefunction. [...] It is non-vanishing only for $\textbf{r}=0$. [...] $\delta(\textbf{r})$ is an eigenfunction of the position operator with ...
0
votes
2answers
70 views

When generalizing from discrete (but infinite) eigenstates to continuous eigenstates, Why do we change the definition?

The propagator function for discrete eigenstates is $$u(t)=\sum_{n=1}^{\infty}|E_n\rangle\langle E_n|e^{-iE_nt/ \hbar } \tag{1}\ .$$ But when we have continuous eigenstates, (like for the case of ...
3
votes
1answer
119 views

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 – ...
0
votes
1answer
414 views

Quantum mechanics, operator commutes with Hamiltonian

My textbook said, if an operator $\hat{O}$ commutes with the Hamiltonian, then we can use the eigen vectors of the Hamiltonian as a basis of the Hilbert space, then express the operator $\hat{O}$ in ...
-1
votes
2answers
125 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
votes
1answer
96 views

Why tensor product? [duplicate]

Let $A$ an $B$ be two discrete observables (like spins). When exactly and why we have to consider their tensor product when talking about the mutual observation of the corresponding phenomena?
4
votes
2answers
163 views

How can mean value of a quantity $be$ an operator?

In Laundau & Lifshitz Quantum Mechanics. Non-relativistic theory in $\S29$ a problem is given: PROBLEM Average the tensor $n_in_k-\frac13\delta_{ik}$ (where $\mathbf{n}$ is a unit vector along ...
2
votes
3answers
128 views

Expression of density operator

States in Quantum Mechanics can be thought of as density operators, i.e., positive semi-definite, normalized trace class operators on a Hilbert Space $\mathcal{H}$. In the case ...
4
votes
2answers
1k views

Derivative of the product of operators and Derivative of exponential

I'm asked to show that $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$ With $\lambda$ a continuous parameter. Should I use the ...
2
votes
1answer
44 views

Connection between half and whole integer eigenvalues for orbital angular momentum [duplicate]

I have been trying to follow this derivation from Sakurai and Shankar, pulling from both. I would like to see how the following derivation can be extended to orbital angular momentum, and thus find ...
5
votes
2answers
396 views

Can expectation value be imaginary?

I was solving a problem and the result of the expectation value of an operator came out to be $-\frac{\hbar}{4}$ $i$. Is this result possible? It seems counter intuitive.
-2
votes
2answers
37 views

Find the eigenvalues of the operator [closed]

A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator. In this question is it necessary to define what the projection operator is? And ...
3
votes
3answers
100 views

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 ...
0
votes
1answer
35 views

Mutually Commutative

What is the definition of a Mutually Commutative set of operators? I've found articles describing a complete set of mutually commutative operators, but I can't actually find what mutually commutative ...
1
vote
1answer
54 views

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?
0
votes
3answers
126 views

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 ...
0
votes
2answers
176 views

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 & ...
0
votes
1answer
40 views

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 ...
0
votes
0answers
56 views

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 ...
0
votes
2answers
50 views

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 $t$ ...
1
vote
2answers
136 views

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 ...
0
votes
0answers
41 views

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). ...
0
votes
0answers
29 views

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 ...
2
votes
2answers
135 views

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 $$A\psi(x)=x\frac{\partial }{\partial x}\psi(x),$$ ...
1
vote
1answer
38 views

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>}, ...
0
votes
1answer
68 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 ...
0
votes
1answer
108 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> ...
0
votes
0answers
31 views

Commutator for time?

I know that in quantum mechanics, we can define space as the operator $\hat{x}=i\hbar \frac{d}{dp}$ in momentum space,and that position does not commute with momentum. However, in general relativity, ...
2
votes
1answer
77 views

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) = ...
1
vote
2answers
59 views

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 ...
3
votes
2answers
146 views

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 ...
3
votes
2answers
107 views

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 ...
0
votes
1answer
38 views

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 ...
1
vote
1answer
40 views

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 ...
1
vote
2answers
54 views

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 ...
2
votes
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 ...
0
votes
1answer
70 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 ...
-2
votes
1answer
52 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 $$Tr\left[e^Ae^B\right]=Tr\left[e^A\right]Tr\left[e^B\right]~?$$ ...
1
vote
0answers
77 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 ...
4
votes
2answers
117 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
votes
0answers
98 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
vote
2answers
94 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 ...