Questions tagged [operators]

In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

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18 views

Why is $\langle \hat{A} \rangle = tr(\hat{\rho} \hat{A})$? [duplicate]

Given that $$\langle \hat{A} \rangle = \langle \psi|\hat{A}|\psi \rangle$$ Why does $\langle \hat{A} \rangle = \mathrm{tr}(\hat{\rho} \hat{A})$, where $\hat{\rho}$ is the density operator, $\hat{\rho} ...
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Proof of the canonical commutator relationship from $\hat{p}=-i\hbar \nabla$

Given that $\hat{r} \psi = r\psi$ where $r$ is the position of a quantum particle, and where $\hat{p}=-i\hbar \nabla$, the notes I have simply state that $$[\hat{r}_i, \hat{p}_j] = i\hbar \delta_{ij}$$...
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How the matrix representation of a Hamiltonian affects the eigenvalues?

Suppose we're given the following Hamiltonian: $$\hat{H}=\frac{\omega}{\hbar} \left(\hat{S}_+^2+\hat{S}_-^2\right)$$ Suppose also that we measure $\vec{S}^2$ and get $6\hbar^2$, i.e. reduced to the $s=...
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Phase of a symmetry

Let's say I have a Hamiltonian $H$ and symmetry of this Hamiltonian such that $[ H, S ]= 0$ It's easy to see that any opertor $\tilde{S} = e^{i\theta} S$ is also a symmetry of $H$ as $[ H, \tilde{S} ]=...
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Exponential of ladder operators acting on a Fock state

I'm trying to evaluate $(\hat{a}+\hat{a}^\dagger)^k|n\rangle$ Where $\hat{a}$ and $\hat{a}^\dagger$ are ladder operators and $|n\rangle$ the $n$th Fock state. For this, I separated the problem in ...
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Quantum Mechanics Spectral decomposition misunderstanding

My notes state that the spectral decomposition formula is of the form: $$ \hat{A} = \hat{A}\hat{1} = \sum{\hat{A} } |A_i\rangle\langle A_i | = \sum{A_i } |A_i\rangle\langle A_i | $$ Now consider the ...
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Transformation operator in Sequence of linear QND measurements

I am following the book Braginsky and Khalili. Consider a measurement scheme where we connect a object to be measured to another quantum system which is then measured by classical devices.(Example: ...
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Understanding the derivation of the partial trace

Consider the derivation as follows: Note that the capital letters are in the Hilbert space B whilst the lower-case letters are in Hulbert space A. I am unsure about the first line of mathematics and ...
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The Negative Energy in the Harmonic Oscillator Potential!

I'm self studying Quantum mechanics from Griffiths. Now I'm at the Harmonic oscillator potential. All my questions raised after defining the ladder operators $a_-$ and $a_+$. If $\psi$ satisfies the ...
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Eigenvalue and eigenfunction of Hamiltonian on interval $[0,1]$

I am seeking the minimum eigenvalue and the eigenfunction of the following Hamiltonian on interval $[0,1]$; $$\hat{P}^2+ c \hat{Q}^{-1} + c (I-\hat{Q})^{-1}$$ where $c $ is an arbitrary positive ...
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Understanding bra-ket notation

So I am a newbie to QM, and coming from math, I believe I am not understanding some key points in bra-ket notation. So given a quantum state $\psi$, I understand that $|\psi \rangle$ is a just a ...
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Why is that in quantum mechanics quite often the physical observable is represented by Hermitian operator? [duplicate]

My knowledge in vector space and quantum mechanics is weak and I am trying to understand and make sense of the question that I asked. It will be very helpful if someone could explain it to me in a ...
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1answer
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Matrix representation of the $\mathfrak{su}(1,1)$ $K$ operators

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\...
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Normal ordering in string theory

So, in QFT where the calculation involves noncommuting operators, we define normal ordering to remove the ambiguity. This is usually done by moving annihilation operators to the right of the creation ...
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1answer
37 views

Relationship between symmetries and quantum operators of classical quantities?

I noticed this the other day. I don't really know "what" this means, I'd love to understand. The energy operator is $\hat E = -i \hbar \frac{\partial}{\partial t}$. Conservation of energy ...
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Matrix element and expectation value

Can I say that the expectation value of an observable $𝐴̂$ for a state $|𝛼⟩$: $⟨𝐴⟩≡⟨𝛼|𝐴̂|𝛼⟩$ is a more general case of the matrix element$⟨𝛼|𝐴̂|\beta⟩$? I'm not quite clear how are they ...
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Commutator of $p$ and $x^n$ [closed]

I found this calculation about the commutator $[p, x^n]$: In lines 4 to 5, it seems like they take an $x$ from $px$ and move it to the left to make $x \cdot x = x^2$. Is that legal? Wouldn't that ...
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In quantum mechanics how the expression of average value of an observable is derived?

In Dirac's Principles of QM following is stated: $$ \langle x | A + B | x \rangle = \langle x | A | x \rangle + \langle x | B |x \rangle $$ but $$ \langle x | AB | x \rangle \ne \langle x | A | x \...
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Confusion about conserved current in quantum field theory

In classical mechanics, it is known from Noether's theorem every continuous symmetry gives a conserved current \begin{equation} \partial_{\mu}J^{\mu}=0, \end{equation} where $J^{\mu}$ (generally) can ...
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Commutators with Hamiltonian of the form $H=\frac{p^2}{2m} + V(x)$

Consider a one-dimensional problem with a Hamiltonian \begin{equation*} H=\frac{p^2}{2m} + V(x) \end{equation*} where $x$ and $p$ are the position and momentum operators, $m$ is the mass and $V(x)$ ...
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$n$th roots of unity as eigenvalues of Hermitian matrix

Recently,my professor told me that for a qudit system we can consider the generalized $\hat{\sigma}_{z}$ observable as $$\hat{Z}=\sum_{k=0}^{d-1}\omega^{k}|k><k|,$$ where $\omega=e^{\frac{2i\pi}{...
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Time ordering operator if commutator is $c$-number function

I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by $$ U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
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51 views

What does it mean to say that the operators evolve in time in the Heisenberg picture?

I get that in the Schrödinger picture the wave function evolve in time and the quantum operators are independent of time. However, in the Heisenberg picture the operators evolve in time and the wave ...
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Units of $\hat{a}$ and $\hat{a}^\dagger$ in discrete vs continuous $k$ and normalization

Consider the quantization of the electromagnetic field. In the discrete case, given in Wikipedia, the operators $\hat{a}$ and $\hat{a}^\dagger$ are dimensionless $$[\hat{a}]=[\hat{a}^\dagger] = 1,$$ ...
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Expectation value of time-independent operators

I'm reading from a lecture note on introductory quantum mechanics (here), which says on P.3 that "The expectation value of any time-independent operator $\hat{Q}$ on a stationary state is time-...
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1answer
59 views

How does the Weyl transform take into account which quasiprobability distribution was used?

I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
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42 views

Velocity operator for Hamiltonian that satisfying generalized Schrodinger equation

The velocity operator is defined as $\mathbf{v}=\frac{i}{\hbar}[H,\mathbf{r}]$ for the Hamiltonian $H$ satisfying $H\psi=\epsilon \psi$. What's the expression of $\mathbf{v}$ for $H$ satisfying the ...
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35 views

Why does the star product satisfy the “Bopp Shift relations”: $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?

In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts": $$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
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Product of time ordered exponential

The following expectation of a time ordered exponential is easy to work with: $$ \left\langle n \right| T \left\lbrace e^{-i \int_{t_1}^{t_2} V_A(t’)dt’ }\right\rbrace T \left\lbrace e^{-i \int_{...
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Time-evolution operator written through a commutator [duplicate]

I found this expression for the time-evolution operator: $$\begin{split} U(t) & = T_{\leftarrow}\exp\left[-i\int_0^t ds H(s)\right] \\ &= \exp\left[-\frac{1}{2}\int_0^t ds\int_0^t ds' [H(s),H(...
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42 views

The definition of the weaker notion of symmetry in the sense of Wigner's theorem

The weaker notion of symmetry, in the sense of Wigner's theorem, is a transformation on the states that leave all quantum mechanical amplitudes invariant. This tells that such transformations are ...
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1answer
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The Eigenstates of a Symmetric Operator

Good Afternoon, By definition, an observable $O$ for a system of N identical particles is symmetric just in case $\langle\psi|O|\psi\rangle = \langle\psi|P^{\dagger}OP|\psi\rangle$ for any permutation ...
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Is the tensor product structure $\hat{H}_0 = \hat{h}_0 \otimes \mathbb{I} + \mathbb{I} \otimes \hat{h}_0$ wrong when interactions are included?

First, consider two uncoupled harmonic oscillators $x_1(t)$ and $x_2(t)$ with classical Lagrangian $$ L_0 = \frac{1}{2} m_1 \dot{x}_1^2 - \frac{1}{2} m_1 \omega_1^2 x_1^2 + \frac{1}{2} m_2 \dot{x}_2^...
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How to Rigorously Find the Explicit Form of the Eigenvalues Equation?

Even in the simplest case of a free particle, so subjected to the Hamiltonian: $$H=\frac{\hat{p}^2}{2m}$$ we often need to find the explicit form of the eigenvalue equation: $$H|E\rangle=E|E\rangle \ \...
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Are there Feynman diagrams for dimension-6 operator?

This document https://arxiv.org/abs/1008.4884 presents in Tables 2 and 3 the mathematical expression of many dimension-6 operators, for example (just an example) The mathematical expression does not ...
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1answer
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Correlation functions - Polchinski equation 6.2.18

At some point of Polchinski book, we are interested in calculate the following correlation function: $$\left\langle \prod_{j=1}^n[e^{ik_i\cdot X(z_i,\bar{z}_i)}]_r\prod_{j=1}^p\partial X^{\mu_j}(z_j')...
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Variance becomes negative [closed]

Say I've an operator $\hat \sigma_{11} = |1\rangle \langle1|$. Now say we calculate the variance of $\hat \sigma_{11}$. So we have, $$\langle (\Delta\hat \sigma_{11})^2 \rangle = \langle \hat \sigma_{...
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Pauli equation: hermite adjoint when deriving probability density

When trying to derive the probability density from the Pauli equation, I face a problem. Starting from the Pauli equation $$ i\hbar \frac{\partial \Psi}{\partial t}=\hat H_0 \Psi +\mu_B \ \hat \sigma \...
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1answer
88 views

Why do we represent states vectors with ket vectors?

From what I currently understand given a general state vector $|\psi\rangle$ the wave function: $$\psi(x) = \langle x|\psi\rangle$$ represent the vector $|\psi\rangle$ in the base of the eigenvalues ...
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Normalisation after applying an operator to a wavefunction

When we apply an operator to a wave function sometimes the result will not be normalised. What does this imply about both the operator and the function? Specifically, when I apply the creation ...
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1answer
63 views

Kochen-Specker theorem for infinite-dimensional Hilbert space

I would like to understand the mathematical content of Kochen-Specker theorem. This theorem states the following: If the dimension of a Hilbert space $\mathcal{H}$ is $>2$ then there is no ...
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3answers
106 views

About the behavior of the position and momentum operators

Following my book I came to know the following expressions for the position and momentum operators ($\hat{x},\hat{p}$): \begin{align}&\langle x|\hat{x}|\psi\rangle=x\psi(x) \ \ \ \ \ &(1)\\[1....
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Regarding Dirac equation heuristic formulation

So in the Wikipedia Page of Dirac equation we are presented with this equation $$\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}...
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In the Schrödinger equation, is there an operator associated with $V$ as there is with $T$ (in the Hamiltonian)?

In the Schrödinger equation we can see an operator associated with the position This operator is used in the expression for the kinetic energy $T$, being part of the quantum mechanical Hamiltonian. ...
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How to apply anti-unitary symmetry operators?

It is know that the symmetry operators can be applied to operators like $$ \hat{O} \stackrel{g}{\rightarrow} \widehat{g O} $$ demand the matrix element to be invariant under symmetry, we have $$ \...
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1answer
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Plugging a global phase into an operator

Cheers to everyone. I' ve got a serious doubt about the following: consider the annihilation operator $\hat a$. For practical reasons, I sometimes find useful redefining it in the following way : $\...
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31 views

Why does quantum analogue of current density operator $\mathbf j = \rho \mathbf v$ involve an anti-commutator?

I am reading through Luttinger's "Theory of Thermal Transport Coefficients". In equation (1.14), the charge-density operator is defined as $$\rho(\mathbf r) = e\sum_j \delta(\mathbf r-\...
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55 views

Invariance of differential operators

How do we prove that del operator is invariant under any kind of change of coordinates, specifically under galilean transformations? I am getting an extra term containing the relative velocity of two ...
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How to find the unitary operation of a depolarizing channel? [duplicate]

Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
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1answer
79 views

Functions versus Vectors in Quantum Mechanics

In the beginning quantum mechanics is introduced by representing the states as cute little complex vectors, for example: $$|a\rangle=a_+|a_+\rangle+a_-|a_-\rangle$$ this is a complex vector ...

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