For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant.
2
votes
1answer
17 views
Left-ratio and right-ratio in (not necessarily commutative) field
I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$.
The authors use terms right-ratio and left-ratio in section 13.
Right-ratio is denoted as $[x_1, x_2, ... ...
2
votes
0answers
30 views
To construct a Schrödinger wave with prescribed mean position and momentum
Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
7
votes
1answer
241 views
Do these two sets of matrices form groups?
Stimulated by some Physics backgrounds, consider the following two sets of matrices.
Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
3
votes
2answers
114 views
Can the $0$-norm represent determinism?
In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm.
Call $\{v_1,\ldots,v_N\}$ a unit vector ...
1
vote
1answer
44 views
Dense in the special unitary group
I was going through the PhD thesis of Daniel Nagaj. In the last paragraph on page 30, there is this following sentence.
A universal gate set must be dense in the group $SU(n)\ldots$
My question ...
0
votes
0answers
17 views
QFT basics for klein gordon fields [migrated]
Hi I am teaching myself QFT from Peskin for next years maths course and I have two questions:
1) What is a c-number is it a complex number and if so why does it mean, ...
1
vote
0answers
29 views
Demonstrate basic property of Hermitian
I want to demonstrate that $(A|v\rangle)^* = \langle v|A^*$
Given only this $(AB)^*=A^*B^*$ and $(|v\rangle, A|w\rangle) = (A^*|v\rangle, |w\rangle)$
Is it possible?
The difficulty is that i don't ...
1
vote
0answers
77 views
Is quantum game theory reducible to classical game theory?
Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
Superposed initial states,
Quantum ...
3
votes
3answers
208 views
Quantum Mathematics?
As per my last question, this has less to do with cold, hard, and fast calculations and more to do with the interplay between mathematics and philosophy...but as armchair philosophers aren't as hard ...
0
votes
0answers
19 views
Circuit identities HTH
Using this circuit indetities $HXH=Z, HYH=-Y, HZH = X$ prove $HTH=R_x(\pi/4)$. here $H$ is Hadamard matrix, $X,Y$ and $Z$ are Pauli matrix, $R_x$ is a rotation matrix and $T=\left[ \begin{array}{cc}
1 ...
1
vote
0answers
21 views
Pure Phase Number
I am read a solution (4.9) Here say:
... both $a, d$ are pure phases, so that it is always possible to find (non unique) real numbers $\alpha, \beta, \delta$ such that $a = ...
0
votes
1answer
27 views
Wave-function with boundary conditions
I'm given a complex wave-function $\psi(t, x)$, in one spatial dimension, which satisfies $i \partial_t \psi = \partial_x^2 \psi$, a simplified form of Schrodinger's equation in one spatial dimension. ...
0
votes
2answers
100 views
Quantum uncertainty can explain the Riemann Hypothesis?
In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
0
votes
2answers
89 views
Bessel function recursion relation
I'm reading a paper and the following set of radial equations is derived:
$
-i \lbrack \partial_r + \frac{1}{r} \left( \frac{1}{2} - \nu \right) \rbrack u(r) = \pm k v(r)
$
$
-i \lbrack \partial_r ...
1
vote
0answers
49 views
At large times, $\sin(\omega t)$ tends to zero?
While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
1
vote
1answer
18 views
Expectation value of pure state in quantum mechanics
It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
0
votes
0answers
14 views
Opposite vector state in Bloch Sphere
I will like understand geometrically Why the opposite vector coordinate $\theta$ of $|\psi\rangle = \cos (\theta/2)|0\rangle + \exp(i\phi)\sin(\theta/2)|1\rangle$ is $\pi - \theta$. Anybody will be ...
2
votes
0answers
39 views
Doubt about the spectrum of an operator
I consider the Laplacian operator
$$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
1
vote
0answers
28 views
Three body problem with point interactions
I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials).
Is ...
1
vote
1answer
56 views
Tensor and Kronecker product
I have a follow exercise. Let $\left|\Psi\right\rangle = \tfrac{1}{\sqrt{2}}\left(\left|0\right\rangle + \left|1\right\rangle\right)$. Write out $\left|\Psi\right\rangle^{\otimes 2}$ xplicitly, both
...
0
votes
0answers
60 views
Translation of an article
I need to read this article
"On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups"
authors:G.Zhislin, A. ...
1
vote
0answers
25 views
References for three body problems with Fermi statistic
I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
3
votes
1answer
77 views
Is the partial trace congruent under a change of basis?
My intuition tells me that the partial trace should be congruent under a change of basis. That is, if I have some matrix $A$ in the space of linear operators acting on a joint hilbert space: $A \in ...
0
votes
1answer
44 views
Dirac's notation? (QM)
I have a question regarding Dirac's notation in quantum physics.
As far as I understand: $\langle a|b\rangle=(a1^*,a2^*)*(b1,b2)^T$
But what does $\langle1/2,1/2|J|1/2,-1/2\rangle$ mean?
1
vote
0answers
46 views
Fourier transform of integral kernel of the free resolvent
The free resolvent in $\mathbb{R}^3$ has this rapresentation
$$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is
...
2
votes
1answer
35 views
General theory behind ladder operators
To derive the representation of SO(3) one uses the ladder operator method. What is the theoretical basis for this method? Often the ladder operators are simply stated in the textbooks of quantum ...
0
votes
0answers
32 views
Solvable Models In Quantum Mechanics
Is there anyone who studied on the book "Solvable Models In Quantum Mechanics" by Albeverio? I don't succed in understanding the proof of page 116 about the eigenvalues of the Hamiltonian with point ...
1
vote
0answers
35 views
Qubit state finding
Suppose we have two qubits in the state $x|00\rangle+y|11\rangle $.
What is the resulting state of the second qubit in that case? Use and to denote and respectively.
2
votes
1answer
34 views
Is multiplying by a measurable function $V$ always self-adjoint?
There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator:
$$-\Delta + V(x)$$
Is (essentially) self-adjoint on ...
3
votes
1answer
61 views
A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$
So I have a strange quantum potential I have been playing with:
$$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$
where $\mu$ is the Möbius function. This is what ...
2
votes
1answer
101 views
Question on complex number calculation for transmission coefficient of finite potential well
This is actually in my quantum mechanics textbook (pure math question though), and I just cannot see why this equality is true. Any help would be greatly appreciated!
Let $F$ and $A$ be nonzero ...
6
votes
1answer
90 views
Kernel of adjoint operator
This problem is puzzling me, even though it should be really simple.
Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
0
votes
0answers
24 views
If $H=\frac{P^2}{2m}+V(X)$, prove $\mathbb{E}_\psi(\frac{P^2}{m})=\mathbb{E}_\psi(XV'(X))$
Suppose we have a quantum mechanical Hamiltonian
$$H=\frac{P^2}{2m}+V(X)$$
How can we show that $$\mathbb{E}_\psi(\frac{P^2}{m})=\mathbb{E}_\psi(XV'(X))$$
Any help would be greatly apprecaited
3
votes
1answer
28 views
Essential selfadjointness preserved under unitarily transfomration?
I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations.
In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an ...
4
votes
5answers
90 views
General solution of $\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\rho^{2}}u$
In his discussion of the radial wave function of hydrogen Griffiths (Introduction to Quantum Mechanics, 2nd ed, p.146) gives the general solution ...
1
vote
2answers
79 views
Show $e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$
How do you show that
$$e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$$
where $\sigma_i$ are the Pauli matrices.
1
vote
0answers
38 views
Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?
Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces:
$$
H = H_1 \otimes \cdots \otimes H_n,
$$
and let $\mathcal{H}$ be a ...
5
votes
0answers
31 views
A non-integrable representation of the Heisenberg Algebra
Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ ...
1
vote
0answers
74 views
Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?
This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
1
vote
0answers
48 views
Convolutions of Path Integrals of Gaussian Functions
I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
0
votes
0answers
50 views
Applying boundary conditions on the general solution of the time-indpendent Schrodinger equation
I'm using the boundary conditions for the general solution of the time-independent Schrodinger equation for a particle in a potential well, and I get the following 2 equations: ...
2
votes
2answers
152 views
Solving the time-independent Schrodinger equation for particle in a potential well
I'm solving a quantum mechanics problem for the particle in a potential well, and the equation I have to solve is $$\frac{d^2\psi}{dx^2}+k\psi=0$$where $$k=\frac{2mE}{\hbar^2}$$
This seems easy enough ...
7
votes
2answers
167 views
“Fun” question: anyone know why $e$ (Euler's Number) was chosen for wave functions?
First, let me say that this is merely something I have always wondered about, and can never seem to find a good reference for. I simply want to know... the geek in me.
Why was $e$ (Euler's Number) ...
1
vote
1answer
32 views
Manipulating derivatives after substitution: $\xi=\gamma x$
I am following a quantum mechanics text book which uses a simple looking substitution in a derivative.
The substitution is $$\xi=\gamma x\tag1$$
It then says that ...
2
votes
2answers
102 views
Is there a reason for the similarity between $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$
I was wondering whether the similarity between the functions $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$ was due to some more fundamental limiting relation between the two functions (or similar ...
1
vote
2answers
57 views
Showing that $\langle p\rangle=\int\limits_{-\infty}^{+\infty}p |a(p)|^2 dp$
How do I show that $$\int \limits_{-\infty}^{+\infty} \Psi^* \left(-i\hbar\frac{\partial \Psi}{\partial x} \right)dx=\int \limits_{-\infty}^{+\infty} p \left|a(p)\right|^2dp\tag1$$
given that ...
1
vote
1answer
39 views
Unwanted $i$ floating around when trying to calculate $\langle p\rangle$
$\def\sp#1{\left\langle#1\right\rangle}$I am given
$$ \Psi(x,0)=A_0 \exp\left(-\frac{x^2}{2\sigma_0^2}\right) \cdot \exp\left(\frac{i}{\hbar}p_0x\right)\tag1$$
where $A_0=(\pi ...
0
votes
0answers
59 views
The general recipe for finding the conjugate of a complex function
I have the general recipe for finding the complex conjugate of a function down as follows:
Suppose I have $f(z)$:
Separate $f(z)$ into a sum of real and imaginary functions such that ...
0
votes
1answer
88 views
Solving time dependant Schrodinger equation in matrix form
If we have a hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector
$$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t),)$$
With Hamiltonian $H$ given by
$$H=\hbar\omega
...
1
vote
1answer
51 views
Two-particle operator in the second quantization
In "Quantum mechanics" by Schwabl I found a chapter (1.3.3) about one- and two-particle operators in the second quantization. The derivation was only sketched and contained this equation:
...
