Tagged Questions
0
votes
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 ...
1
vote
1answer
106 views
Why is this not a realisable operation on a quantum system?
Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
5
votes
2answers
111 views
Quantum Mechanical Operators in the argument of an exponential
In Quantum Optics and Quantum Mechanics, the time evolution operator
$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$
is used quite a lot.
Suppose $t_i =0$ for simplicity, and say the ...
0
votes
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}} = ...
1
vote
1answer
89 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 ...
2
votes
2answers
112 views
How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?
I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times:
$\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue
$W$.
...
1
vote
1answer
47 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 ...
3
votes
2answers
185 views
Coherent State, Unitary Operators, Harmonic Oscillator
Consider the operator:
$$O = e^{\theta(a^\dagger b - b^\dagger a)}$$
where $\theta$ is a constant.
$O$ is a unitary operator.
$a$, $a^\dagger$, $b$, and $b^\dagger$ are ladder operators for two ...
4
votes
2answers
94 views
Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$
I know how to derive below equations found on wikipedia and have done it myselt too:
\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= ...
2
votes
1answer
54 views
The matrix element of a normal-ordered operator
Eq (1.137) in Negele and Orland gives the following identity for a normal-ordered operator $A(a_i^\dagger,a_i)$:
$$\langle \phi|A(a_i^\dagger,a_i)|\phi'\rangle=A(\phi_i^*,\phi'_i)e^{\sum ...
4
votes
2answers
76 views
Translator Operator
In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
0
votes
1answer
85 views
Matrix representation for fermionic annihilation operator
My guess it should look something like this:
$ c_\sigma = ...
3
votes
2answers
117 views
Quantum commutator
I'm given this commutator:
$$\left[PXP,P\right]$$
Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$
I've solved it in two ways, the first one is just aplying the commutator to some function ...
2
votes
1answer
92 views
Hermitian Adjoint of differential operator
I came across this equation (identity) (Eq. 4 in this paper):
$\int(-i d\psi/dx)^*\psi dx = \int \psi^*(-i d\psi/dx) dx + id(\psi^*\psi)/dx\mid_{-\infty}^{+\infty}$
I have trouble proving it. I ...
4
votes
3answers
106 views
Associating a Unitary operator to proper Lorentz transformations?
If one reads eg page 32 of Srednicki where he says:
In quantum theory, symmetries are represented by unitary (or
antiunitary) operators. This means that we associate a unitary
operator U(Λ) ...
2
votes
1answer
58 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
-1
votes
1answer
82 views
Operators in quantum mechanics
According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$?
If so then how?
And if we transfer to integrate formulation then how it will look like?
1
vote
0answers
26 views
Quantum graph theory: complex spectra
In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely ...
3
votes
1answer
106 views
The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
10
votes
3answers
325 views
How to tackle 'dot' product for spin matrices
I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
3
votes
5answers
231 views
Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
2
votes
1answer
170 views
Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $
I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that
$$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
2
votes
1answer
124 views
Coordinate representation of quantum ladder operator?
I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics.
I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
1
vote
2answers
244 views
Deriving a QM expectation value for a square of momentum $\langle p^2 \rangle$
I alredy derived a QM expectation value for ordinary momentum which is:
$$
\langle p \rangle= \int\limits_{-\infty}^{\infty} \overline{\Psi} \left(- i\hbar\frac{d}{dx}\right) \Psi \, d x
$$
And i ...
0
votes
4answers
382 views
Product of exponential of operators
in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators
$$e^{A}e^{B}=e^{A+B} $$
if and only if
$$[A,B]=0$$
I remember my professor told use about ...
4
votes
4answers
135 views
How to prove that the symmetrisation Operator is hermitian?
Let $\mathcal{H}_N$ be the $N$ particle Hilbert space. So a quantum state $\left| \Psi \right>$ may be representated by
$$\left| \Psi \right> = \left| k_1 \right>^{(1)}\left| k_2 ...
1
vote
3answers
149 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.
$$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, ...
5
votes
2answers
170 views
Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
5
votes
4answers
270 views
Is the momentum operator diagonal in position representation?
The matrix elements of the momentum operator in position representation are:
$$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$
Does this imply that $\langle x ...
1
vote
1answer
49 views
Question about the linearity of wave functions
For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
1
vote
1answer
54 views
Can I prove boundedness of an operator without checking it for its whole domain?
(I don't have a direct reference so this is a little fishy and I'll delete it if nobody recognises what I'm talking about, but I though for starters I'll ask anyway)
I've heard at university that if ...
0
votes
0answers
34 views
Why there is no operator for time in QM? [duplicate]
Is there one central reason why there is no "Time" operator in QM?
I know this question has been asked before, but I thought I would try to stimulate some fresh thinking.
1
vote
2answers
240 views
Derivatives of operators
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
0
votes
1answer
132 views
Evaluate Commutator with Partial Derivatives
I need to evaluate the following commutator...
$[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$
i tried applying an ...
3
votes
2answers
143 views
Sum of two density matrices: $\rho=p_1\rho_1+p_2\rho_2$
Suppose we have
$$\rho=p_1\rho_1+p_2\rho_2$$
Where $\rho_1$ and $\rho_2$ are density matrices with $p_1+p_2=1$
I'm trying to show this is also a density matrix
If we let
$$\rho_1=\sum_i^n ...
1
vote
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 ...
2
votes
1answer
114 views
Physical Significance of Operator Norm/Spectral Norm of a Quantum Operator
Is there any physical significance of operator norm/spectral norm of a hermitian operator?
6
votes
1answer
170 views
“An operator is hermitian”. Implications?
Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as:
Every dynamical variable may be ...
4
votes
1answer
189 views
How to evaluate spin operators in second quantization for spin symmetry-broken Slater determinants?
Suppose we have the following Slater determinant:
\begin{equation}
| \Psi \rangle = \prod \limits_{i,i'} a^+_{i\alpha} a^+_{i'\beta} | \rangle
\end{equation}
where $a^+_{i\alpha}$ creates an electron ...
3
votes
3answers
463 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:
$$\hat{p} ...
2
votes
2answers
286 views
Constructing the exponential form of a unitary operator
I think I've got this figured out but wanted to make sure I'm doing this right.
Working with operators that satisfy bosonic commutation relations $[b,b^\dagger] = 1$, I define a very general unitary ...
2
votes
2answers
120 views
What is the nature of the correspondence between unitary operators and reversible change?
Why does the formalism of QM represent reversible changes (eg the time evolution operator, quantum gates, etc) with unitary operators?
To put it another way, can it be shown that unitary ...
2
votes
2answers
135 views
Why must quantum logic gates be linear operators?
Why must quantum logic gates be linear operators? I mean, is it just a consequence of quantum mechanics postulates?
12
votes
1answer
373 views
Intuitive meaning of Hilbert Space formalism
I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:
The observables are given by self-adjoint operators on the ...
4
votes
2answers
227 views
Eigenvalues of a quantum field?
Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t).
In quantum field theory, the classical field is promoted to an operator-valued ...
4
votes
2answers
181 views
When you apply the spin operator, what exactly is does it tell you?
The example I'm trying to understand is:
$ \hat{S}_{x} \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
\end{pmatrix} = 1/2 \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
...
2
votes
1answer
373 views
A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?
I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector:
...
6
votes
1answer
170 views
Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
4
votes
1answer
106 views
Constructing the space of quantum states
I want to learn how to construct spaces of quantum states of systems.
As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is ...
1
vote
1answer
205 views
Adjoint of momentum operator
In position basis, we have,
$$\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 ...
