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.
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 ...
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 ...
7
votes
2answers
168 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) ...
7
votes
1answer
252 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 ...
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 ...
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 ...
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 $X$ ...
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 ...
4
votes
4answers
187 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 ...
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 ...
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
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 ...
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 ...
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
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 ...
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 ...
3
votes
2answers
115 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 ...
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 ...
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 ...
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 ...
3
votes
1answer
79 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 ...
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 ...
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
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
2answers
91 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 ...
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 ...
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 ...
2
votes
1answer
102 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 ...
2
votes
2answers
154 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 ...
2
votes
2answers
174 views
Instance of Ehrenfest's Theorem
Please Help me to fill in the gaps
Show
$$ \frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$
$$\frac{\text d \langle {p} ...
2
votes
1answer
18 views
Left-ratio and right-ratio in (not necessarily commutative) field
I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$.
The authors use terms right-ratio and left-ratio in section 13.
Right-ratio is denoted as $[x_1, x_2, ... ...
2
votes
1answer
36 views
Is multiplying by a measurable function $V$ always self-adjoint?
There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator:
$$-\Delta + V(x)$$
Is (essentially) self-adjoint on ...
2
votes
1answer
260 views
Relationship between dual space and adjoint of a linear operator
I am having a hard time understanding the concept of adjoint of a linear operator. Given a finite dimensional Hilbert space $H$ over a field $F$, I know the dual space is the vector space $H^*$ of all ...
2
votes
1answer
69 views
boundary conditions for Schr$\ddot{\textrm{o}}$dinger equation in 2D polars?
What are the boundary conditions at $r=0$ for Schr$\ddot{\textrm{o}}$dinger's equation for a quantum particle in 2D polars $(r,\theta)$, with potential $U=0$ for $r<a$ and $U=\infty$ for $r>a$?
...
2
votes
1answer
107 views
How are Tr(AB) results restricted?
In quantum mechanics one postulates that for each state $i$ there is a matrix $A_i$ and for each measureable dynamical variable $j$ (velocity etc.) there is a matrix $B_j$. Both are Hermitian matrices ...
2
votes
0answers
30 views
To construct a Schrödinger wave with prescribed mean position and momentum
Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
2
votes
0answers
39 views
Doubt about the spectrum of an operator
I consider the Laplacian operator
$$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
2
votes
0answers
68 views
Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?
A lot of known quantum gates are in the Pauli group $(I, X, Z, Y)$ or in the Clifford group $(H, P, Cnot)$. I need examples of the quantum gates that aren't in this groups. Also, are there are matlab ...
2
votes
0answers
72 views
Defining entanglement in subspaces of tensor product
Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
2
votes
0answers
123 views
How do I solve this integral
How do I solve this integral (expectation value) :
$$\int_{-\infty}^{\infty} \psi (x)^* \hat p \psi (x)\ dx.$$
where the $\hat p =-i\hbar \frac {\partial}{\partial x}$ is an operator and $\psi (x)$ is ...
2
votes
0answers
36 views
Integration problem calculating the density of a mixed state
This is a solved example in a text which I'm being unable to grasp. Sorry if it is overely easy.
Consider a qubit which points in any direction of the space with equal probability. We can simply ...
2
votes
1answer
57 views
Positive Operator Value Measurement Question
I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators $\{E_m\}$ for ...
1
vote
4answers
91 views
Solution to a system of quadratics
I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system
$$\alpha_0^2+\alpha_1^2=1$$
$$\beta_0^2+\beta_1^2=1$$
...
1
vote
2answers
79 views
Show $e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$
How do you show that
$$e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$$
where $\sigma_i$ are the Pauli matrices.
1
vote
1answer
52 views
Two-particle operator in the second quantization
In "Quantum mechanics" by Schwabl I found a chapter (1.3.3) about one- and two-particle operators in the second quantization. The derivation was only sketched and contained this equation:
...
1
vote
1answer
1k views
Qubits and vector projections
In $\Bbb C^2$, how many real unit vectors are there whose projection onto $|1\rangle$ has length $\sqrt{3}/2$?
I would think zero as $\bigl(\frac{\sqrt{3}}{2}\bigr)^2 + x^2 = 1$, therefore there are ...
1
vote
1answer
153 views
Why are the coefficients of the base states of a qubit complex numbers?
Why are qubits represented as $$\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; ...
1
vote
1answer
46 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 ...
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
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 ...
