The operators tag has no wiki summary.
1
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2answers
610 views
Derivative of the product of operators
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 definition
...
1
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2answers
164 views
Proof of $Dq-qD=1$ where $D=\frac{\partial }{\partial q}$ is the differential operator
Can anyone provie me the proof of $Dq-qD=1$ where $D=\frac{\partial }{\partial q}$ refers to the differential operator?
Or if it's something special to quantum mechanics, why is it?
Is this ...
1
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2answers
239 views
Deriving a QM expectation value for a square of momentum $\langle p^2 \rangle$
I alredy derived a QM expectation value for ordinary momentum which is:
$$
\langle p \rangle= \int\limits_{-\infty}^{\infty} \overline{\Psi} \left(- i\hbar\frac{d}{dx}\right) \Psi \, d x
$$
And i ...
1
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2answers
167 views
Physical meaning of some operators formed by $|Q\rangle \langle Q|$
In Dirac's formulation of quantum mechanics,
Suppose that $q$ represents position observable.
About $|q\rangle \langle q|$: what does this operator mean? I do get that it results in an operator, but ...
1
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3answers
120 views
The notion of bounded states in quantum mechanics and their characterization with operators
Is there any case of potential $V$, such that the continuity of the operator
$H=c\ \Delta+V$
is not spoiled?
And I don't know any non-differnetial operator examples for continous spectra. I ...
1
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1answer
46 views
Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$
I just finished deriving the commutators:
\begin{align}
[\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\
[\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\
\end{align}
On the ...
1
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3answers
147 views
Operators explaination and momentum operator in QM
I know and understand why equation below holds. But i am new to operator thing in QM and would need some explaination on this.
$$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, ...
1
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1answer
96 views
Klein-Gordon Canonical Commutation Relation (CCR)
In the complex Klein-Gordon field we regard as dynamical variables the field $\phi$, the complex conjugate of the field $\phi^*$, and the momenta $\pi$, $\pi^*$. I can't see how should arise the ...
1
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1answer
272 views
The Hermiticity of the Laplacian (and other operators)
Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator?
Alternatively: is the matrix representation of the Laplacian Hermitian?
i.e.
$$\langle \nabla^{2} x | y \rangle = \langle x | ...
1
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1answer
88 views
Some Dirac notation explanations
Equation for an expectation value $\langle x \rangle$ is known to me:
\begin{align}
\langle x \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x
\end{align}
By the definition we ...
1
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1answer
54 views
Can I prove boundedness of an operator without checking it for its whole domain?
(I don't have a direct reference so this is a little fishy and I'll delete it if nobody recognises what I'm talking about, but I though for starters I'll ask anyway)
I've heard at university that if ...
1
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1answer
130 views
Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?
I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did ...
1
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1answer
205 views
Adjoint of momentum operator
In position basis, we have,
$$\langle x \mid \hat p \mid \Psi(t) \rangle = -\imath \hbar \frac{\partial{\langle x \mid \Psi(t) \rangle}}{\partial{x}} $$
Now i know $\hat{p}$ is a hermitian operator ...
1
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4answers
206 views
Hamiltonian in position basis
Let $ H = \frac{-h^2}{2m}\frac{\partial^2 }{\partial x^2}$. I want to find the matrix elements of $H$ in position basis. It is written like this:
$\langle x \mid H \mid x' \rangle = ...
1
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1answer
149 views
Inverse of a sum of two easy matrices
Let $A$ be a symmetric positive semidefinite matrix and $I$ the identity matrix.
Given the linear equation
$$
y = A(A + \sigma^2I)^{-1} x
$$
Write $A$ in terms of its eigenvectors $|u_i\rangle$,
...
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5answers
482 views
Operator vs linear transformation
One of the postulates of quantum mechanics is that every physical observable corresponds to a Hermitian operator $H$, that the possible outcomes of the measurements are eigenvalues of the operator, ...
1
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1answer
102 views
Why is this not a realisable operation on a quantum system?
Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
1
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1answer
67 views
Notational techniques for dealing with creation operators on Fock space
This question is trying to see if anyone has some simple notation (or tricks) for dealing with operators acting on coherent states in a Fock space. I use bosons for concreteness; what I'm interested ...
1
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1answer
49 views
Question about the linearity of wave functions
For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
1
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2answers
125 views
Notation for differential operators and wave function math
I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac ...
1
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1answer
161 views
Multiplication of 3-vector operators
I've started reading "Quantum Mechanics: A Modern Development" by Leslie E. Ballentine and have some trouble understanding how to handle 3-vector operators (i.e. an operator $\mathbf{A}$ with ...
1
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0answers
36 views
QFT basics for Klein-Gordon fields
I am teaching myself QFT from Peskin for next years maths course and I have two questions:
What is a c-number? Is it a complex number, and if so why does it mean, ...
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0answers
26 views
Quantum graph theory: complex spectra
In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely ...
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0answers
75 views
Explicit evaluation of a radially ordered product
I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in
...
1
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2answers
240 views
Derivatives of operators
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
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votes
4answers
382 views
Product of exponential of operators
in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators
$$e^{A}e^{B}=e^{A+B} $$
if and only if
$$[A,B]=0$$
I remember my professor told use about ...
0
votes
2answers
185 views
How is an arbitrary operator usually denoted in quantum mechanics?
Which symbols are usually used to denote an arbitrary operator in quantum mechanics, such as O in the following example?
$O \mbox{ is Hermitian} \Leftrightarrow \Im{\left< O \right>} = 0$
0
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1answer
131 views
Evaluate Commutator with Partial Derivatives
I need to evaluate the following commutator...
$[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$
i tried applying an ...
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1answer
562 views
Derivation of angular momentum commutator relations
I'm trying to understand the derivation of the angular momentum commutator relations. How is
$$[zp_y, zp_x] ~=~ 0?$$
How is
$$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$
0
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1answer
39 views
Eigenvalue $a_n$
Q1:
In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
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1answer
84 views
Matrix representation for fermionic annihilation operator
My guess it should look something like this:
$ c_\sigma = ...
0
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1answer
164 views
State normalization in Dirac's formulation of quantum mechanics
Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then
we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $
Our ...
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1answer
106 views
When does a function of an operator act in the same way as the operator?
"Consider an operator $A = r - a$, where $r$ is an operator and $a$ is a constant. Consider only those state kets $V_i$ in the Hilbert space such that $AV_i = 0$ ($A$ acting on $V_i$). Define a ...
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1answer
439 views
Operators and Commutator Definitions
I have several problems with General Definitions of an Operator and Commutator :
the product of operators is generally not commutative:
$$\hat A \hat B \not= \hat B\hat A .$$
what is this means ...
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1answer
319 views
Probability of getting a particular spin
I'm a beginner in quantum mechanics, and I'm a bit confused about states and the probability to measure certain values. I would like to understand at least the following simplified situation:
...
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1answer
318 views
Weird operator and wavefunctions
How can one show that $\int_{-\infty}^{\infty}\psi^*(x)(d/dx+\tanh x)(-d/dx+\tanh x)\psi(x) dx=\int_{-\infty}^{\infty} |(d/dx+\tanh x)\psi(x)|^2 dx$, where $\psi$ is normalized?
0
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0answers
38 views
An application of Toeplitz operators
I want to find an application of the Toeplitz operators. All I need is a known problem (not an open problem) which solution use the theory of Toeplitz operators. I don't need all the details but I ...
0
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0answers
28 views
Schrodinger equation in momentum space [duplicate]
I have a problem this is:
When I solve the Schrodinger equation in momentum space, I had done as this:
$\begin{array}{l}
i\hbar \frac{{\partial \Psi }}{{\partial t}} = - \frac{\hbar ...
0
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0answers
76 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
0
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0answers
38 views
Time ordering and Fermions
Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should
$T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$
return ...
0
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0answers
34 views
Why there is no operator for time in QM? [duplicate]
Is there one central reason why there is no "Time" operator in QM?
I know this question has been asked before, but I thought I would try to stimulate some fresh thinking.
0
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1answer
97 views
Positive Permutation Tensor
I have seen that one can make an operator with
$$
L^i=\epsilon^{ijk}x_{j}\partial_{k}
$$
But what if I want to make instead items that are sums, instead of differences. For instance
...
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1answer
65 views
Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]
I have a problem with the Hamiltonian, I don't think anything to solve it!!
So could you give me some hints!
Knowing that:
$$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
-1
votes
1answer
82 views
Operators in quantum mechanics
According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$?
If so then how?
And if we transfer to integrate formulation then how it will look like?
-1
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1answer
184 views
Properties of expectation values of quantum operators [closed]
$$\langle \hat A \rangle \langle \hat B \rangle=\langle \hat A\hat B \rangle,$$
$$\langle \hat A \rangle + \langle \hat B \rangle=\langle \hat A + \hat B \rangle,$$
$$\langle \hat A^2 \rangle ...
