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.
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
$$H=\hbar\omega
...
1
vote
1answer
52 views
Two-particle operator in the second quantization
In "Quantum mechanics" by Schwabl I found a chapter (1.3.3) about one- and two-particle operators in the second quantization. The derivation was only sketched and contained this equation:
...
0
votes
1answer
47 views
Inequality from Von Neumann entropy.
I am looking over some old course notes. First, Von Neumann entropy is defined.
The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
0
votes
1answer
171 views
Show that $A^{\dagger^{\dagger}} = A $
How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that ...
2
votes
2answers
174 views
Instance of Ehrenfest's Theorem
Please Help me to fill in the gaps
Show
$$ \frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$
$$\frac{\text d \langle {p} ...
2
votes
1answer
260 views
Relationship between dual space and adjoint of a linear operator
I am having a hard time understanding the concept of adjoint of a linear operator. Given a finite dimensional Hilbert space $H$ over a field $F$, I know the dual space is the vector space $H^*$ of all ...
1
vote
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 \\
...
3
votes
2answers
68 views
Mathematical explanation of problems behind time and space derivatives being second order
$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$
with the wave function $\phi$ being a relativistic scalar: a complex
number which has ...
1
vote
1answer
68 views
substitution in a non linear differential equation and to get a nicer form
well I had this equation at the begining
$$
i \frac{\partial u}{\partial{z}} + \frac{1}{2 k_0} \frac{\partial^2 u}{\partial x^2} +\frac{1}{2}k_0 n_1 F(z) x^2 u-\frac{i[g(z) -\alpha(z)]}{2}u + k_0 ...
2
votes
2answers
188 views
“proof” A is a Hermitian Matrix
For an arbitrary complex matrix A show that $$A*A^\dagger$$ is Hermitian.
Where the dagger "$\dagger$" stands for the "complex conjugate and transpose" operators.
From what I understand this must ...
2
votes
0answers
68 views
Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?
A lot of known quantum gates are in the Pauli group $(I, X, Z, Y)$ or in the Clifford group $(H, P, Cnot)$. I need examples of the quantum gates that aren't in this groups. Also, are there are matlab ...
3
votes
5answers
403 views
Hydrogen atom in partial differential equations
For the hydrogen atom, if
$$\int |u|^2 ~dx = 1,$$
at $t = 0$,
I am trying to show that this is true at all later times.
What I need help is with differentiating the integral with respect to $t$, and ...
2
votes
0answers
72 views
Defining entanglement in subspaces of tensor product
Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
4
votes
1answer
70 views
Quantum Information: Deutsch-Jozsa Algorithm
There is a step in the construction of this algorithm which I'm not understanding:
$\displaystyle \left[\sum_x \frac{| x \rangle}{\sqrt{2^n}}\right]\left[\frac{ | 0 \rangle -| 1 \rangle ...
3
votes
1answer
144 views
'Quantum' approach to classical probability
Quantum mechanics defines a state of a system as a superposition of 'classical' states with complex coefficients, thus reducing many problems to linear algebra. Can classical statistics be approached ...
3
votes
1answer
87 views
What are the requirements for a “test” function to show operators commute?
To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$.
My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...
4
votes
4answers
188 views
Applications of Operator Algebras to modern physics
I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
1k views
Qubits and vector projections
In $\Bbb C^2$, how many real unit vectors are there whose projection onto $|1\rangle$ has length $\sqrt{3}/2$?
I would think zero as $\bigl(\frac{\sqrt{3}}{2}\bigr)^2 + x^2 = 1$, therefore there are ...
0
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 ...
10
votes
1answer
283 views
Quantization of angular momentum: is Dirac's proof wrong?
I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
2
votes
2answers
92 views
Perturbation theorem of Weyl
Does anyone know where to find something about the perturbation theorem of Weyl, preferably
on the internet. The theorem I'm talking about states:
let $A$ be a self-adjoint operator on a Hilbert ...
5
votes
0answers
241 views
Studying quantum mechanics without physics background
I am a first year PhD math student, and I am wondering if I should study quantum mechanics even though I don't have an undergrad background in physics.
I posted this question in physics ...
10
votes
1answer
171 views
Does this notion of morphism of noncommutative rings appear in the ring theory literature?
Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
1
vote
1answer
119 views
Under What Conditions Does the Action of the Dual Space Induce an Hermitian Inner Product?
I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space ...
1
vote
4answers
91 views
Solution to a system of quadratics
I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system
$$\alpha_0^2+\alpha_1^2=1$$
$$\beta_0^2+\beta_1^2=1$$
...
2
votes
0answers
123 views
How do I solve this integral
How do I solve this integral (expectation value) :
$$\int_{-\infty}^{\infty} \psi (x)^* \hat p \psi (x)\ dx.$$
where the $\hat p =-i\hbar \frac {\partial}{\partial x}$ is an operator and $\psi (x)$ is ...
2
votes
0answers
36 views
Integration problem calculating the density of a mixed state
This is a solved example in a text which I'm being unable to grasp. Sorry if it is overely easy.
Consider a qubit which points in any direction of the space with equal probability. We can simply ...
3
votes
1answer
80 views
Function space in QM
I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to ...
0
votes
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}
...
2
votes
1answer
57 views
Positive Operator Value Measurement Question
I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators $\{E_m\}$ for ...
1
vote
0answers
290 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 ...
2
votes
1answer
69 views
boundary conditions for Schr$\ddot{\textrm{o}}$dinger equation in 2D polars?
What are the boundary conditions at $r=0$ for Schr$\ddot{\textrm{o}}$dinger's equation for a quantum particle in 2D polars $(r,\theta)$, with potential $U=0$ for $r<a$ and $U=\infty$ for $r>a$?
...
5
votes
1answer
169 views
Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.
I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces.
I was wondering what is the physical ...
0
votes
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 ...
3
votes
1answer
128 views
Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow P_{j}P_{i}=0$
I am reading Introduction to Quantum Computing by Kaye, Laflamme, and Mosca. As an exercise, they write
"Prove that if the operators $P_{i}$ satisfy $P_{i}^{*}=P_{i}$ and $P_{i}^{2}=P_{i}$ , then ...
2
votes
1answer
107 views
How are Tr(AB) results restricted?
In quantum mechanics one postulates that for each state $i$ there is a matrix $A_i$ and for each measureable dynamical variable $j$ (velocity etc.) there is a matrix $B_j$. Both are Hermitian matrices ...
0
votes
1answer
102 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)]$$
of the complex Ginzburg - Landau equation:
...
0
votes
1answer
121 views
Solving $-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon)$
I am trying to solve this differential equation:
$$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$
This was found ...
1
vote
1answer
153 views
Why are the coefficients of the base states of a qubit complex numbers?
Why are qubits represented as $$\left|{q}\right\rangle = \alpha\left|{0}\right\rangle+\beta\left|{1}\right\rangle\equiv\alpha\left[{1 \ 0}\right]^T+\beta\left[{0 \ 1}\right]^T; ...
3
votes
2answers
249 views
Regarding Ladder Operators and Quantum Harmonic Oscillators
When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator:
Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
