Questions tagged [quantum-mechanics]
For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.
1,728
questions
1
vote
1
answer
49
views
Why is the modulus of a function taken before it is squared?
I am a studying quantum mechanics, and frequently I see that the modulus of a function has been taken before it is squared.
For example, in a problem I was working on, the following was in the ...
- 141
0
votes
1
answer
31
views
Commutator on a finite-dimensional $\mathcal{H}$ is scalar multiple of an identity
Suppose $A$ and $B$ are operators on a finite-dimensional Hilbert space and suppose that $[A, B] = c I$ for some constant c. Show that $c = 0$.
I have tried approaching the problem using matrices $A, ...
1
vote
1
answer
61
views
Moller Operator and the Determination of Bound States in Quantum Scattering Theory
I am trying to understand the Moller operator in quantum scattering theory. An important result concerning this operator is the following useful property that can be used to determine bound states.
...
- 854
2
votes
0
answers
33
views
Nuclear spaces: the Schwartz class
Good morning, I'm studying quantum mechanics as a mathematician.
I read that the Schwartz class
$$\begin{equation*}
\mathcal{S}(\mathbb{R}) := \{ \varphi \in \mathcal{C}^\infty (\mathbb{R}, \...
- 61
1
vote
0
answers
31
views
Hardy Hille type eigenfunction exapansion
I am trying a figure out Eigen-function expansion of the following kind.
$$
\exp\left[{-\frac{1}{4\alpha} \left(x^2 + y^2\right)}\right] I_{l + \frac{1}{2}} \left(\frac{x y}{2 \alpha}\right) = \sum_{n}...
- 65
2
votes
1
answer
109
views
Mathematically motivated derivation of Feynman path integral
In Hall's book Quantum Theory for Mathematicians he gives a wonderful derivation of the Feynman path integral. His formal derivation is as follows.
Let $\psi \in L^2(\mathbb{R}^n)$ and $\hat{H}$ the ...
- 6,255
0
votes
0
answers
12
views
Continuity on countably-normed Hilbert spaces
i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an ...
- 61
0
votes
0
answers
35
views
Shooting method for a couple of non linear ODE representing Helium atom
In quantum mechanics we can maybe express the s-states (spherically symmetric wave functions) of the Helium atom as two wave function depending on the spherical radius variable $r$ ($0 < r < + \...
- 191
0
votes
0
answers
30
views
Deriving Components of a Quaternion-Based Wave Function
I am currently exploring an intriguing topic related to quaternion-based wave functions and have encountered a mathematical challenge that I hope to get some insights on. The concept is detailed in an ...
- 1
0
votes
0
answers
26
views
Lossless steering Order of Magnitude estimation
This is an equation from Rau's book "Quantum theory-an Information Processing Approach". It's leading to "lossless steering" of a particle if you insert an infinite number of ...
1
vote
0
answers
44
views
Sum of Spherical Harmonics and Rotational Invariance
PROBLEM
Suppose that
$$
\sum_{m=-l}^{l} c_m Y_m^l(\theta, \phi) Y_m^l(\theta', \phi')^*
$$
is rotationally invariant, then how can we show that the $c_m$'s must be all equal?
ATTEMPT AT A SOLUTION
I ...
- 1,179
2
votes
1
answer
42
views
Help With Showing Local Integrability of Inverse Fourier Transform (Quantum Theory for Mathematicians Chapter 4 Problem 1 Part b)
I am working on part (b) of exercise 1 in Chapter 4 of Brian C. Hall's book. It states:
$\textbf{Exercise 1}$:
A $\textbf{locally integrable}$ function $\psi(x,t)$ satisfies the free Schr$\ddot{\...
- 353
1
vote
2
answers
54
views
Hint for showing the identity $(\nabla\psi\cdot\nabla)\nabla\psi^*+(\nabla\psi^*\cdot\nabla)\nabla\psi=\nabla|\nabla \psi|^2$
I need some help to show this following identity. $$(\nabla\psi\cdot\nabla)\nabla\psi^*+(\nabla\psi^*\cdot\nabla)\nabla\psi=\nabla|\nabla \psi|^2$$
My attempt:
$$
\partial_i\psi\partial_i\partial_k\...
- 151
1
vote
1
answer
61
views
Is true that $F(H)\otimes F(H)\cong F(H)$? where $F(H)$ is the full fock space of Hilbert space $H$
Let $H$ be a Hilbert space. Define the full fock space $F(H)=\bigoplus\limits_{n=0}^\infty H^{\otimes n}$ where $H^{\otimes 0}=\Bbb{C}\Omega$, $\Omega$ is an element of $H$ of unit norm. Let us define ...
- 1,130
2
votes
1
answer
46
views
A proof of no-cloning theorem in the case of pures states in a qubit system
I am working on this problem
Consider a qubit $\scr H =\Bbb C^2$ and pure states. Prove the no-clonning theorem
( hint: Use the linearity of the channel to arrive to a contradiction)
I wonder if the ...
- 589