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 $\times$, $+$!
2,746
questions
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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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0answers
16 views
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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1answer
33 views
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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0answers
25 views
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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0answers
32 views
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 ...
1
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1answer
40 views
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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1answer
3 views
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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0answers
17 views
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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2answers
45 views
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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0answers
33 views
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 ...
1
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2answers
83 views
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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1answer
37 views
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 ...
1
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1answer
44 views
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 $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\...
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30 views
Normal ordering in string theory
So, in QFT where the calculation involves noncommuting operators, we define normal ordering to remove the ambiguity. This is usually done by moving annihilation operators to the right of the creation ...
1
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1answer
37 views
Relationship between symmetries and quantum operators of classical quantities?
I noticed this the other day. I don't really know "what" this means, I'd love to understand.
The energy operator is $\hat E = -i \hbar \frac{\partial}{\partial t}$. Conservation of energy ...
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1answer
47 views
Matrix element and expectation value
Can I say that the expectation value of an observable $š“Ģ$ for a state $|š¼ā©$: $āØš“ā©ā”āØš¼|š“Ģ|š¼ā©$ is a more general case of the matrix element$āØš¼|š“Ģ|\betaā©$? I'm not quite clear how are they ...
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1answer
51 views
Commutator of $p$ and $x^n$ [closed]
I found this calculation about the commutator $[p, x^n]$:
In lines 4 to 5, it seems like they take an $x$ from $px$ and move it to the left to make $x \cdot x = x^2$. Is that legal? Wouldn't that ...
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2answers
58 views
In quantum mechanics how the expression of average value of an observable is derived?
In Dirac's Principles of QM following is stated:
$$
\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 \...
2
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3answers
73 views
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 ...
1
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1answer
71 views
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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1answer
46 views
$n$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}{...
5
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2answers
45 views
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
$$
U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
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1answer
51 views
What does it mean to say that the operators evolve in time in the Heisenberg picture?
I get that in the Schrƶdinger picture the wave function evolve in time and the quantum operators are independent of time. However, in the Heisenberg picture the operators evolve in time and the wave ...
0
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1answer
66 views
Units of $\hat{a}$ and $\hat{a}^\dagger$ in discrete vs continuous $k$ and normalization
Consider the quantization of the electromagnetic field. In the discrete case, given in Wikipedia, the operators $\hat{a}$ and $\hat{a}^\dagger$ are dimensionless $$[\hat{a}]=[\hat{a}^\dagger] = 1,$$ ...
0
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1answer
26 views
Expectation value of time-independent operators
I'm reading from a lecture note on introductory quantum mechanics (here), which says on P.3 that "The expectation value of any time-independent operator $\hat{Q}$ on a stationary state is time-...
0
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1answer
59 views
How does the Weyl transform take into account which quasiprobability distribution was used?
I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation ...
0
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0answers
42 views
Velocity operator for Hamiltonian that satisfying generalized Schrodinger equation
The velocity operator is defined as $\mathbf{v}=\frac{i}{\hbar}[H,\mathbf{r}]$ for the Hamiltonian $H$ satisfying $H\psi=\epsilon \psi$. What's the expression of $\mathbf{v}$ for $H$ satisfying the ...
1
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1answer
35 views
Why does the star product satisfy the “Bopp Shift relations”: $f(x,p)\star g(x,p)=f(x+\frac{i}{2}\partial_p,p-\frac{i}{2}\partial_x) g(x,p)$?
In (Curtright, Fairlie, Zachos 2014), the authors mention (Eq. (14) in this online version) the following relation, known as "Bopp shifts":
$$f(x,p)\star g(x,p)=f\left(x+\frac{i}{2}\...
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0answers
54 views
Product of time ordered exponential
The following expectation of a time ordered exponential is easy to work with:
$$ \left\langle n \right| T \left\lbrace e^{-i \int_{t_1}^{t_2} V_A(tā)dtā }\right\rbrace T \left\lbrace e^{-i \int_{...
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0answers
67 views
Time-evolution operator written through a commutator [duplicate]
I found this expression for the time-evolution operator:
$$\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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2answers
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 ...
1
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1answer
35 views
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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0answers
71 views
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
$$
L_0 = \frac{1}{2} m_1 \dot{x}_1^2 - \frac{1}{2} m_1 \omega_1^2 x_1^2 + \frac{1}{2} m_2 \dot{x}_2^...
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1answer
71 views
How to Rigorously Find the Explicit Form of the Eigenvalues Equation?
Even in the simplest case of a free particle, so subjected to the Hamiltonian:
$$H=\frac{\hat{p}^2}{2m}$$
we often need to find the explicit form of the eigenvalue equation:
$$H|E\rangle=E|E\rangle \ \...
0
votes
1answer
58 views
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 ...
2
votes
1answer
56 views
Correlation functions - Polchinski equation 6.2.18
At some point of Polchinski book, we are interested in calculate the following correlation function: $$\left\langle \prod_{j=1}^n[e^{ik_i\cdot X(z_i,\bar{z}_i)}]_r\prod_{j=1}^p\partial X^{\mu_j}(z_j')...
0
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1answer
47 views
Variance becomes negative [closed]
Say I've an operator $\hat \sigma_{11} = |1\rangle \langle1|$. Now say we calculate the variance of $\hat \sigma_{11}$. So we have,
$$\langle (\Delta\hat \sigma_{11})^2 \rangle = \langle \hat \sigma_{...
0
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2answers
28 views
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
$$ i\hbar \frac{\partial \Psi}{\partial t}=\hat H_0 \Psi +\mu_B \ \hat \sigma \...
1
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1answer
88 views
Why do we represent states vectors with ket vectors?
From what I currently understand given a general state vector $|\psi\rangle$ the wave function:
$$\psi(x) = \langle x|\psi\rangle$$
represent the vector $|\psi\rangle$ in the base of the eigenvalues ...
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0answers
70 views
Normalisation after applying an operator to a wavefunction
When we apply an operator to a wave function sometimes the result will not be normalised. What does this imply about both the operator and the function?
Specifically, when I apply the creation ...
1
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1answer
63 views
Kochen-Specker theorem for infinite-dimensional Hilbert space
I would like to understand the mathematical content of Kochen-Specker theorem. This theorem states the following:
If the dimension of a Hilbert space $\mathcal{H}$ is $>2$ then there is no ...
0
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3answers
106 views
About the behavior of the position and momentum operators
Following my book I came to know the following expressions for the position and momentum operators ($\hat{x},\hat{p}$):
\begin{align}&\langle x|\hat{x}|\psi\rangle=x\psi(x) \ \ \ \ \ &(1)\\[1....
0
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1answer
52 views
Regarding Dirac equation heuristic formulation
So in the Wikipedia Page of Dirac equation we are presented with this equation
$$\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}...
2
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3answers
93 views
In the Schrödinger equation, is there an operator associated with $V$ as there is with $T$ (in the Hamiltonian)?
In the Schrƶdinger 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.
...
2
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0answers
31 views
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
$$
\...
2
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1answer
49 views
Plugging a global phase into an operator
Cheers to everyone. I' ve got a serious doubt about the following: consider the annihilation operator $\hat a$. For practical reasons, I sometimes find useful redefining it in the following way : $\...
0
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2answers
31 views
Why does quantum analogue of current density operator $\mathbf j = \rho \mathbf v$ involve an anti-commutator?
I am reading through Luttinger's "Theory of Thermal Transport Coefficients". In equation (1.14), the charge-density operator is defined as
$$\rho(\mathbf r) = e\sum_j \delta(\mathbf r-\...
1
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2answers
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 ...
1
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0answers
16 views
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\...
3
votes
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 ...
