The operators tag has no wiki summary.
2
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5answers
3k views
Mathematical background for Quantum Mechanics
What are some good sources to learn the mathematical background of Quantum Mechanics?
I am talking functional analysis, operator theory etc etc...
6
votes
2answers
331 views
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
3
votes
2answers
509 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, ...
12
votes
1answer
372 views
Intuitive meaning of Hilbert Space formalism
I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:
The observables are given by self-adjoint operators on the ...
3
votes
1answer
171 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 ...
3
votes
1answer
157 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):
...
0
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0answers
74 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}} = ...
10
votes
2answers
889 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 ...
8
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2answers
499 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 ...
1
vote
6answers
407 views
Is H=H* sloppy notation or really just incorrect, for Hermitian operators?
I saw it in this pdf, where they state that
$P=P^\dagger$ and thus $P$ is hermitian.
I find this notation confusing, because an operator A is Hermitian if
$\langle \Psi | A \Psi \rangle=\langle A ...
4
votes
1answer
624 views
Evolution operator for time-dependent Hamiltonian
When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
...
1
vote
2answers
609 views
Derivative of the product of operators
I'm asked to show that
$\frac{d(\hat{A}\hat{B})}{d\lambda} = \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$
With $\lambda$ a continuous parameter
Should I use the definition
...
0
votes
1answer
164 views
State normalization in Dirac's formulation of quantum mechanics
Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then
we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $
Our ...
10
votes
3answers
322 views
How to tackle 'dot' product for spin matrices
I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
6
votes
3answers
243 views
What is the physical meaning of weak expectation values?
In the two-state formalism of Yakir Aharanov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. ...
5
votes
2answers
164 views
Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
3
votes
3answers
461 views
Matrix elements of momentum operator in position representation
I have two related questions on the representation of the momentum operator in the position basis.
The action of the momentum operator on a wave function is to derive it:
$$\hat{p} ...
3
votes
1answer
707 views
Why/How is this Wick's theorem?
Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.
$$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$
...
3
votes
2answers
212 views
Is there a four dimensional form of Born's Rule -redub
Generalizing Born's Rule for 4-dimensions $x_4$, write
$$\langle a\rangle = \int\Psi A\Psi^* \mathrm{d}x_4$$
Is this consistent with quantum mechanics?
Is this a generalized form of the Born's ...
2
votes
2answers
104 views
Matrix representing the quantity - why can some matrices not be physical quantity?
In Heisenberg picture, my textbook says that the following matrix
$A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity.
the book says this is because ...
1
vote
5answers
480 views
Operator vs linear transformation
One of the postulates of quantum mechanics is that every physical observable corresponds to a Hermitian operator $H$, that the possible outcomes of the measurements are eigenvalues of the operator, ...
6
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2answers
694 views
Difficulties with bra-ket notation
I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
4
votes
1answer
190 views
Existence of adjoint of an antilinear operator, time reversal
The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a ...
3
votes
2answers
161 views
Derivative of a Position Eigenket
I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that ...
3
votes
1answer
46 views
Linearizing Quantum Operators [duplicate]
Possible Duplicate:
Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ ...
2
votes
1answer
56 views
A physical quantity that is a real combination and commutability
Suppose that a matrix
$$A ~=~ x_1 B + x_2 C$$
is a linear combination of two self-adjoint matrices $B$ and $C$.
I'm interested in when $A$ represents a physical quantity.
When the linear ...
1
vote
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 ...
0
votes
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
votes
4answers
379 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 ...
