The operators tag has no wiki summary.
0
votes
1answer
108 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 ...
1
vote
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 $|u_i\rangle$,
...
0
votes
1answer
442 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 ...
0
votes
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
...
4
votes
1answer
122 views
Eigenvalue of $L_z$
In section 4.3 of Griffths' "Introduction to Quantum Mechanics", just below Figure 4.6, the sentence begins
Let $\hbar \ell$ be the eigenvalue of $L_z$ at this top rung...
Why is this valid? ...
2
votes
1answer
203 views
How could $\textbf{S}^2$ not be a multiple of the identity?
I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$:
As will be shown in ...
0
votes
1answer
320 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:
...
3
votes
1answer
707 views
Why/How is this Wick's theorem?
Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.
$$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$
...
4
votes
1answer
226 views
How do we measure $i[\hat\phi(x),\hat\phi(y)]$ in QFT?
What operational procedure is required to measure $i[\hat\phi(x),\hat\phi(y)]$ in an interacting (or non-interacting) QFT? [assume smearing by test-functions, or give an answer in Fourier space, for ...
5
votes
2answers
582 views
Basic Question - Green's Functions in Quantum Mechanics
I am trying to learn about Green's functions as part of my graduate studies and have a rather basic question about them:
In my maths textbooks and a lot of places online, the basic Greens function G ...
3
votes
1answer
158 views
Wick Order and Radial Ordering in CFT
I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case):
...
1
vote
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 ...
4
votes
0answers
67 views
Shape of the state space under different tensor products
I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).
Recall: In a ...
0
votes
1answer
563 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
$$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$
2
votes
1answer
137 views
A missing factor of 2 in the standard Hartree-Fock mean field?
Let's start from a very simple argument: If $A$ and $B$ are some operators, then I can write their product as
$$AB = (A-\langle A\rangle)(B - \langle B \rangle) + \langle A \rangle B + A \langle B ...
4
votes
1answer
129 views
The difference between projection operators and field operators in QFT?
Is there a good reference for the distinction between projection operators in QFT, with an eigenvalue spectrum of $\{1,0\}$, representing yes/no measurements, the prototype of which is the Vacuum ...
7
votes
1answer
75 views
Representation on Hilbert space of the product of two symmetry transformations
We know by Wigner's theorem that the representation of a symmetry transformation on the Hilbert space is either unitary and linear, or anti-unitary and anti-linear.
Let $T$ and $S$ be two symmetry ...
3
votes
4answers
201 views
How to apply an algebraic operator expression to a ket found in Dirac's QM book?
I've been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac's book. In the fourth edition, 33rd page, starting from this:$$\xi|\xi'\rangle=\xi'|\xi'\rangle$$
(Where ...
3
votes
1answer
299 views
Can we solve the particle in an infinite well in QM using creation and annihilation operators?
The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can ...
5
votes
3answers
313 views
Some questions on observables in QM
1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable?
2-What are the criteria to say whether ...
4
votes
1answer
861 views
Compatible Observables
My QM book says that when two observables are compatible, then the order in which we carry out measurements is irrelevant.
When you carry out a measurement corresponding to an operator $A$, the ...
3
votes
0answers
41 views
Spectrum of a quantum relativistic “distance squared” operator
This question disusses the same concepts as that question (this time in quantum context). Consider a relativistic system in spacetime dimension $D$. Poincare symmetry yields the conserved charges $M$ ...
8
votes
2answers
500 views
Regularisation of infinite-dimensional determinants
Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM?
Edit: I failed to make myself clear. In finite ...
3
votes
3answers
572 views
Index Manipulation and Angular Momentum Commutator Relations
I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed.
We have the relations
$$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$
...
1
vote
5answers
482 views
Operator vs linear transformation
One of the postulates of quantum mechanics is that every physical observable corresponds to a Hermitian operator $H$, that the possible outcomes of the measurements are eigenvalues of the operator, ...
2
votes
1answer
313 views
Question on ladder operators
Suppose we have a finite , discrete set of orthonormal states $|k\rangle $
We can construct raising and lowering operators intuitively, for example $$a_+ =\sum_{k=1}^nC_{k+1}|k+1\rangle \langle k|$$
...
3
votes
2answers
389 views
Why is the Dirac operator so important - in both physics and mathematics?
Why is the Dirac operator considered so important - in both physics and (pure) mathematics?
12
votes
4answers
576 views
Energy is actually the momentum in the direction of time?
By comparatively examining the operators
a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?
3
votes
2answers
510 views
Hermitian operator and reality of eigenvalues
Prove or disprove:
The eigenvalues of an operator are all real if and only if the operator is hermitian.
I know the proof in one way; that is, I know how to prove that if the operator is hermitian, ...
3
votes
1answer
46 views
Linearizing Quantum Operators [duplicate]
Possible Duplicate:
Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ ...
3
votes
1answer
172 views
Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$
The right hand side ...
4
votes
4answers
383 views
Unitary Operator as a complex valued function
A book on Quantum Mechanics by Schwinger states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator."
Please give a hint on how to prove this assertion.
0
votes
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?
2
votes
1answer
161 views
Degeneracy and the Hamiltonian
How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) ...
1
vote
2answers
611 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
...
11
votes
1answer
85 views
Metric interpretation of self-adjoint extensions?
I am wondering if beyond physical interpretation, the one dimensional contact interactions (self-adjoint extensions of the the free Hamiltonian when defined everywhere except at the origin) have a ...
30
votes
2answers
214 views
Physical interpretation of different selfadjoint extensions
Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
2
votes
1answer
181 views
Expectation of a commutation relation
Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.
5
votes
1answer
343 views
Simultaneously commuting set
How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these ...
2
votes
2answers
368 views
Operator relation involving the logarithm of an operator?
Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?
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?
$O \mbox{ is Hermitian} \Leftrightarrow \Im{\left< O \right>} = 0$
3
votes
2answers
212 views
Is there a four dimensional form of Born's Rule -redub
Generalizing Born's Rule for 4-dimensions $x_4$, write
$$\langle a\rangle = \int\Psi A\Psi^* \mathrm{d}x_4$$
Is this consistent with quantum mechanics?
Is this a generalized form of the Born's ...
3
votes
2answers
515 views
Commutators and Hermiticity - Exam question
I'm doing old exam questions, and here is one that on first glance seemed rather simple to me, but I just can't get it:
Given are two operators $A$ and $B$, and all we know about them is that
...
4
votes
4answers
577 views
How to calculate the quantum expectation of frequency of a particle?
I know how to calculate the expectation of < $\Psi$|A|$\Psi$ >
where the operator A is the eigenfunction of energy, momentum or position, but I'm not sure how to perform this for a pure frequency.
...
10
votes
2answers
890 views
Applications of the Spectral Theorem to Quantum Mechanics
I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
2
votes
5answers
3k views
Mathematical background for Quantum Mechanics [duplicate]
What are some good sources to learn the mathematical background of Quantum Mechanics?
I am talking functional analysis, operator theory etc etc...
