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.
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1answer
26 views
Expecation for tensor products
We are told that $$Z|0 \rangle = | 0\rangle \\ Z | 1\rangle =-|1\rangle \\ X|0\rangle =|1\rangle \\ X|1\rangle =|0\rangle$$
and we have the state $$|\psi\rangle =|0\rangle |1\rangle +|1\rangle ...
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1answer
18 views
Expectation value of pure state in quantum mechanics
It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
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1answer
58 views
Tensor and Kronecker product
I have a follow exercise. Let $\left|\Psi\right\rangle = \tfrac{1}{\sqrt{2}}\left(\left|0\right\rangle + \left|1\right\rangle\right)$. Write out $\left|\Psi\right\rangle^{\otimes 2}$ xplicitly, both
...
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1answer
39 views
Unwanted $i$ floating around when trying to calculate $\langle p\rangle$
$\def\sp#1{\left\langle#1\right\rangle}$I am given
$$ \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$$
where $A_0=(\pi ...
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1answer
131 views
Commutator relationship proof $[A,B^2] = 2B[A,B]$
I'm trying to find the condition necessary for this commutator relationship equality:
$$[A,B^2]=2B[A,B]$$
So far I've done this:
\begin{align*}
[A,B^2] & = B[A,B] + [A,B]B \\
...
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1answer
68 views
substitution in a non linear differential equation and to get a nicer form
well I had this equation at the begining
$$
i \frac{\partial u}{\partial{z}} + \frac{1}{2 k_0} \frac{\partial^2 u}{\partial x^2} +\frac{1}{2}k_0 n_1 F(z) x^2 u-\frac{i[g(z) -\alpha(z)]}{2}u + k_0 ...
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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 ...
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0answers
80 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 ...
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0answers
24 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 = ...
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0answers
51 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 ...
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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 ...
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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?
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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
...
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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.
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0answers
40 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 ...
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0answers
77 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 ...
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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 ...
1
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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 ...
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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 ...
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0answers
292 views
Exponential of an operator
The definition of the exponential of an operator is given by the following relation:
\begin{equation}
e^U\equiv\sum_{n=0}^\infty\frac{U^n}{n!}
\end{equation}
This definition is a relation in the ...
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1answer
121 views
Solving $-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon)$
I am trying to solve this differential equation:
$$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$
This was found ...
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1answer
175 views
Show that $A^{\dagger^{\dagger}} = A $
How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that ...
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1answer
48 views
Inequality from Von Neumann entropy.
I am looking over some old course notes. First, Von Neumann entropy is defined.
The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
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1answer
66 views
Partial trace of a system with isolated evolution
Let $\rho_{AB}$ be the state of a composite quantum system with state space $H_A\otimes H_B$ (two finite dimensional Hilbert spaces). Now assume that $A$ and $B$ are isolated and suffer a unitary ...
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2answers
828 views
Re-writing in sign basis.
$\newcommand\ket[1]{\left\vert #1\right\rangle}$ Let $\ket\phi = 12 \ket{0} + 1 + 2\sqrt{i2}\ket{1}$. Write $\ket\phi$ in the form $\alpha_0\ket{+} + \alpha_1\ket{-}$. What is $\alpha_0$?
I came ...
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1answer
89 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
$$H=\hbar\omega
...
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1answer
239 views
Vector space generated by the tensor products of pauli matrices
Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices:
\begin{equation}
...
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1answer
262 views
triple integral (quantum mechanics)
I recently started a quantum mechanics course after a long time with no serious maths and I'm having some problems with the most basic maths operations.
Please, help me solve this triple integral ...
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1answer
103 views
Solution of the complex Ginzburg - Landau equation
Can someone show that it's possible to find a solution of the kind:
$$\Phi(x,t)=R(x,t)\exp[i\Psi(x,t)]$$
of the complex Ginzburg - Landau equation:
...
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0answers
24 views
Proving that this is not a positive operator
Let $\rho$ be a density operator (i.e., it is an ortho projection with rank one, and also a positive operator).
Say $X = X^*$ with a spectral decomposition $X = 1P_1 + 4P_4 + 16P_{16}$, and $Y = Y^*$ ...
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0answers
10 views
Can Hessian matrix of probability density function be called density matrix for quantum mechanic
how to calculate density matrix from view of probability for quantum mechanic
Hessian matrix is positive definite, can it be density matrix?
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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, ...
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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 ...
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1answer
29 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. ...
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2answers
100 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 ...
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0answers
15 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 ...
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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. ...
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1answer
47 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?
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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 ...
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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
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votes
0answers
54 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: ...
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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 ...
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votes
0answers
37 views
Vector projections in real units [duplicate]
Possible Duplicate:
Qubits and vector projections
In $\mathbb{C}^2$, how many real (unit) vectors are there whose projection onto $|1\rangle$ has length $\frac{3}{\sqrt{2}}$?
I would think ...
-1
votes
3answers
121 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 ...
