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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Apparent spacetime dependence of creation and annihilation operators

I'm currently going through an introduction to QFT I've stumbled upon. Having trouble with equation (2.29), I'm asking the question: Do creation and annihilation operators ($\hat a^\dagger(k)$ and ...
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1answer
34 views

Time derivative of $\hat p$ under time varying Hamiltonian

How does one show that $$ \frac {d\hat p}{dt} = \frac 1{i\hbar}[\hat p, \hat H]$$ is valid even when Hamiltonian is time dependent explicitly? I can see that this is true when $\hat H$ is time ...
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31 views

Angular Momentum Operators confusion [on hold]

I asked a question earlier that now realize sort of made no sense, but that is only because I was combining a few things in my head....this is the question I meant to ask. How would you calculate ...
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2answers
114 views

Interpreting some domain issues of (potential) momentum operators

In the context of mathematical quantum mechanics, a well known no-go theorem known as Hellinger-Töplitz tells us that an unbounded, symmetric operator cannot be defined everywhere on the Hilbert space ...
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1answer
48 views

Proof of inability to obtain global phase

I'm curious if there's a quick proof of the inability to obtain global phase from a quantum state, since they're supposedly indistinguisable. I suppose to measure this, you would need a Hermitian ...
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47 views

Problem with commutator [closed]

My area is mathematics; however, told me about a problem that can solve with simple math, it says: Using $[\hat{J_{z}},\hat{O}]$, show that $\langle\alpha,j,m|\hat{O}|\beta,j',m'\rangle$ can be ...
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63 views

How to compute the normal ordered angular momentum of a Klein-Gordon real scalar in terms of ladder operators?

I'm trying to compute the angular momentum $$Q_i=-2\epsilon_{ijk}\int{d^3x}\,x^kT^{0j}\tag{1}$$ where ...
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33 views

Is it sensible to speak of the parity operator in 4 dimensional Hilbert space?

So I'm dealing with a system of two qubits, with the hamiltonian given by $$H = \left[ \begin{array}{cccc} a & 0 & -b & 0 \\ 0 & 0 & 0 & -b\\ -b & 0 & 0 & 0\\ 0 ...
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2answers
224 views

Position operator in QFT

My Professor in QFT did a move which I cannot follow: Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
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1answer
52 views

Finding the spectrum of a curious hamiltonian

I wish to analyse the following hamiltonian, i.e. find its eigenvalues and eigenstates. $$H = \frac{1}{2}\epsilon(\sigma _z \otimes \mathbb{1} + 1\otimes \sigma _z) - \Delta (\sigma _x \otimes \sigma ...
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Momentum operator in Dirac formalism

Could you derive the momentum operator as follows: Since $\mathcal{T}(\Delta x)=\exp(-ip_{x} \Delta x/ \hslash)$, if we set $\Delta x=x-0$ then it follows that $\left \langle x\right | ...
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1answer
71 views

Where does the partial derivative come from in Sakurai's derivation of the momentum operator?

How is the momentum operator derived in Dirac formalism? I am reading Quantum Mechanics by Sakurai and he gives the following derivation. But I don't understand how he goes from the third equation to ...
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1answer
39 views

Defining creation and annihilation operators

Creation and annihilation operators can be defined in several different ways, some more general than others. We usually choose to denote by $a$ the annihilation operator and by $a^\dagger$ the ...
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684 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' ...
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1answer
28 views

Constructing matrix for spin in Stern-Gerlach experiment for arbitrary angle

This is a conceptual question about a problem in Sakurai. I understand how to solve the problem, but there's something about it that irks me, and it feels like I'm missing something. In the problem, ...
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2answers
67 views

Eigenvalues being physical observables

I think I'm comfortable with the PDE solutions to the Schrodinger equation. But as soon as we start putting these values in a matrix (in dirac notation), I lose my understanding and everything ...
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2answers
158 views

Grassmann number representation for fermions

How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place? $$ ...
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2answers
77 views

Domain of simple quantum harmonic oscillator

When discussing the spectral theory of unbounded operators, one often starts with an operator defined on a densely defined subspace of your Hilbert space, and then proves that the operator is ...
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Split property for type III algebras entails practical separability

I am reading Halvorson's thesis (http://philsci-archive.pitt.edu/346/1/main-new.pdf), however I don't understand a proof at p.50 where he tries to explain why the split property allows a local agent ...
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1answer
72 views

Momentum Representation vs Position Representation

We are given an operator $g$ from $\mathcal{l}^2(\mathbb{Z})$ to $\mathcal{l}^2(\mathbb{Z})$, i.e., the space of functions that are square summable over $\mathbb{Z}$ such that ...
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2answers
560 views

Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
3
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1answer
43 views

Kraus operator rank

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $\mathcal{d}$ can be generated by an operator-sum representation containing at most $\mathcal{d^2}$ elements. Extending ...
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3answers
191 views

How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$

Wikipedia indicates that the following relation is "easily shown": $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$, however I'm having some trouble showing it. I think I'm just ...
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3answers
82 views

Intuition on positive-operator valued measures (POVM)

I'm having a little trouble understanding what positive-operator valued measure (POVM) are- in particular why/how they are non-negative. For instance, if they just represent measurements, what about ...
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1answer
73 views

Determine eigeinvalue and eigenvector of two operators R and L [closed]

Question: Let H be a Hilbert space with countable-infinite orthonormal basis ${|n>}_{n \in N}$. The two operators R and L on H are defined by their action on the basis elements \begin{align} ...
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1answer
28 views

Translational operator on potential

In https://wiki.oulu.fi/download/attachments/14553161/lattice.pdf I have a problem with the translational operator: The second line under the first figure says $$\tau^\dagger(a)V(x)\tau(a)=V(x+a).$$ ...
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0answers
72 views

An Operator Identity relating to Trace [duplicate]

Suppose that $\hat H$ is an operator (typically a Hamiltonian) and $\beta$ is a positive parameter (typically $\beta=1/k_BT$). Show that $$ \mathbf{Tr}\Big(e^{-\beta\hat H}\Big) \geq ...
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1answer
65 views

Does there exist a state for which $\Delta\sigma_x^2=\Delta\sigma_y^2=0$? If not, how does one prove it?

I just realized that the uncertainty principle says that $$\Delta\sigma_x^2 \Delta\sigma_y^2 \ge \left(\overline{\hat\sigma_z}\right)^2,$$ where ...
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3answers
174 views

Classical Hamiltonian involving product of factors whose quantum analogues don't commute

Dirac remarked in his quantum mechanics book: One can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the ...
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2answers
343 views

Where to place the operator?

I believe it's standard to place the operator in between the conjugate of the wavefunction and the wavefunction itself. For instance, $$\langle p\rangle = \int_{-\infty}^{\infty}\Psi * ...
3
votes
2answers
104 views

Why is $\hat{p} \circ \hat{p}$ the operator corresponding to $p^2$?

I understand from several heuristic arguments that in one dimension, the quantum-mechanical operator $\hat{p} = -i\hbar\,\partial_x$ corresponds to the classical momentum $p$, in the sense that a ...
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1answer
66 views

Uncertainty in position and kinetic energy

How do you find the uncertainties for $x$ and $K$? Knowing that the general uncertainties = $$ \sigma_A \sigma_B \geq 1/2\int \psi ^*[\hat A,\hat B] \psi dx\, $$ I figured out the commutator, for ...
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0answers
34 views

Where Does the Exponent Come From in the Expression for the Rotation Operator

I am currently reading John S. Townsend's "A Modern Approach to Quantum Mechanics." In section 2.2 he introduces the $\hat J$ operator, which he refers to as "the generator of rotations." He gives the ...
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2answers
99 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$>$ ...
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1answer
232 views

Evaluate $\langle \mathbf{p} | 1/\hat{r} | \mathbf{p}' \rangle$

In Sakurai's Problem 1.27 b), we use $\langle \mathbf{r} | \mathbf{p}\rangle = e^{i\mathbf{p}\cdot\mathbf{r}/\hbar}$ to show that $$ \langle \mathbf{p} | F(\hat{r}) | \mathbf{p}' \rangle = ...
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2answers
47 views

How does Dirac conclude that $X_r(c_r)$ cannot vanish?

On page 32 of Dirac's book Principles of Quantum Mechanics, he considers the case when the linear, Hermitian$^1$ operator $\xi$ satisfies an algebraic equation $$\phi(\xi)\equiv(\xi - c_1)(\xi - ...
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1answer
54 views

How to prove that if the expectation value of $A$ in any state is real, then $A$ is Hermitian?

If the expectation value of operator $A$ in any state is real, then $A$ is Hermitian. there is an uncompleted proof: $$ \int(c_1\psi_1+c_2\psi_2)^* A (c_1\psi_1+c_2\psi_2)dx$$ ...
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Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
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1answer
99 views

How does Dirac form this conjugate imaginary equation?

On page 30 of Dirac's book $$\xi|P\rangle = a|P\rangle\tag{12}$$ He then says Suppose we have a solution of (12) and we form the conjugate imaginary equation, which will read $$\langle ...
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1answer
60 views

Expectation value of number operator $\hat{n}$

I'm studying for my quantum mechanics test and I've stumbled on this problem. They want the expectation value of $\hat{n}$, $\langle \hat{n} \rangle$, with this given $\psi$ at $t=0$: $$ \lvert ...
3
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2answers
194 views

Measuring non-commuting observable at once

Given an Hilbert space $H$ (finite dimensional for sake of clarity), and two non-commuting operators $$A = \sum_a a |a\rangle\langle a|$$ and $$B=\sum_a b |b\rangle\langle b|,$$ is it possible to find ...
0
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1answer
60 views

Commutativity of Position Operators

Does the position operator $q_{i}$ of one harmonic oscillator commute with the position operator $q_{j}$ of another different harmonic oscillator? In other words, is $q_{i} q_{j} = q_{j} q_{i}$ true? ...
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1answer
109 views

Time evolution of a quantum system

A quantum system has Hamiltonian $H$ with normalised eigenstates $\psi_n$ and corresponding energies $E_n$ ($n = 1,2,3...$). A linear operator $Q$ is defined by its action on these states: $$ ...
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0answers
18 views

How is it shown that the composition of two real operators is generally not real? [duplicate]

Dirac on page 28 of his QM book writes: Thus the conjugate complex of the product of two linear operators equals the product of the conjugate complexes of the factors in the reverse order. As ...
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1answer
51 views

Why does the raising and lowering operator not affect total angular momentum?

My notes define: $$ L_{\pm} = L_{x} \pm i L_{y} $$ and states: $$ [L_{z},L_{\pm}] = \pm \hbar L_{\pm} $$ I'm fine with this as it's easy to show the result with some ugly algebra. It then says: ...
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1answer
29 views

What are absolutely continuous spectrum and singularly continuous spectrum?

I am now reading some mathematical note on Anderson localization. It mentioned two types of continuous spectrum. What are absolutely continuous spectrum and singularly continuous spectrum? I only had ...
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1answer
48 views

Mapping Issues with Unbounded Operators

Consider the operator-valued generalized function $\phi^{(k)}_{l}:=\phi^{(k)}_{l}$ on space-time $\mathcal{M}$. Now, smooth the operator-valued generalized function with test function $f(x)$ ...
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2answers
57 views

What is the commutator of the exponential derivative operator and the exponential position operator?

What is the commutator of the exponential derivative operator and the exponential position operator? \begin{align} \left[\exp(\partial_x),\exp(x)\right] = \exp(\partial_x)\exp(x) - ...
4
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1answer
163 views

Directional derivatives in the multivariable Taylor expansion of the translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
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1answer
111 views

Creation and Annihilation Operators

Let $\widehat{a}^{+}_{i}$ and $\widehat{a}_{i}$ be the usual bosonic creation and annihilation operators. Consider $$\widehat{q}_{i} = \sqrt{\frac{\hbar}{2m_{i}w_{i}}}(\widehat{a}_{i}+ ...