Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.

Quantum mechanics, also known as quantum physics or quantum theory, deals with classical physical phenomena at quantum scales.

The precise nature of the subject has changed over the years. This article explains its current formulation.

Quantum mechanics aims toprovide a mathematical description of the dual particle-like and wave-like behaviors and interactions of energy and matter.

1699 questions
2
votes
1 answer

If the multiplication of two operators is hermitian, then will they commute?

It is proven that if two operators $\hat{X}$ and $\hat{Y}$ commute, then the multiplication of them will be hermitian, i.e. if $\hat{X}\hat{Y}=\hat{Y}\hat{X}$, then $\left(\hat{X}\hat{Y}\right)^\dagger=\hat{X}\hat{Y}$. My question is that is the…
farhad mabrooki
  • 23
2
votes
1 answer

Explicit Matrix Form of the Time-Evolution Operator

I have been given a time-dependent Hamiltonian $H = \eta$ cos $\omega t$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$ and asked to calculate explicitly in matrix form the time-evolution operator $U(0, t)$ associated to $H$. I am completely…
B Delamera
  • 183
  • 1
  • 9
2
votes
1 answer

Missing term in path integral calculation?

Following Takhtajan (Quantum Mechanics for Mathematicians, AMS 2008, chp. 5), I am trying to calculate the propagator associated with the one-dimensional quantum harmonic oscillator. At one point, it becomes necessary to simplify the expression $$S…
giobrach
  • 7,490
2
votes
2 answers

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 $$\Psi(x)=\frac{1}{\sqrt{2 \pi \hbar}}\int…
Joebevo
  • 1,439
2
votes
3 answers

Instance of Ehrenfest's Theorem

Please Help me to fill in the gaps Show $$ \frac{\text d \langle {p} \rangle}{ \text{d} t} =\left\langle - \frac{ \partial V }{\partial x} \right\rangle .$$ $$\frac{\text d \langle {p} \rangle}{ \text{d} t} $$ $$= \frac{\text d}{\text d t}…
Elements In Space
  • 1,987
2
votes
1 answer

Ladder operators and Quantum Harmonic Oscillators

This question refers to a harmonic oscillator with length parameter a. In terms of raising and lowering operators, the operator $\hat{p}_x^{4}$ for such an oscillator can be expressed as $\hat{p}_x^4 = \frac{\hbar^4}{4a^4}(\hat{A}-\hat{A^†})^4$ (a)…
Mike A
  • 377
2
votes
2 answers

If 2d is only length and width, but not depth then can something really exist in 2d?

2d is length and width, not depth. So if I have an object in 2d (0 depth), then it would be non-existent, right? Paper, graphite, etc. have depth even if it is extremely small. Here is my own theory on the 2cnd dimension:The first thing that comes…
Emanuel Van Rijn
  • 21
2
votes
1 answer

Help With the Proof of the Spectral Decomposition (Quantum Mechanics)

$\newcommand\dag\dagger$ I need to present a proof of the spectral decomposition and I need help in some parts. I will state the theorem and the proof indicating where help is needed. I know this is more mathematical than physics but I encountered…
Pablo
  • 33
1
vote
1 answer

Wavefunction of electron above grounded conductor

Consider a non-relativistic electron moving above a large, flat grounded conductor while it is attracted by its image charge, but cannot penetrate the conductor's surface. What is the Hamiltonian of the electorn and the BC its wavefunction must…
user44212
  • 53
1
vote
1 answer

Positivity and Complete positivity of Simon Map

Simon map in a specific basis is defined as $$ \left[ {\begin{array}{ccc} A & B & C \\ D & E & F \\ G & H & I \\ \end{array} } \right] \rightarrow \left[ {\begin{array}{ccc} A +E & -B & -C \\ -D & E+I& -F \\ -G & -H & I+A …
Kishor Bharti
  • 193
1
vote
2 answers

Problem with a wavefunction in Quantum Mechanics (math) (Book solution possibly wrong?)

Well there is a problem in my book which lists this problem: Calculate the probability that a particle will be found at $0.49L$ and $0.51L$ in a box of length $L$ when it has (a) $n = 1$. Take the wave function to be constant in this range. The…
Biochemist_HK
  • 13
1
vote
1 answer

Eigenfunctions of Hamiltonian in 'natural units'

Let $H=0.5(p^2+q^2)$ be the Hamiltonian in natural units. Let $f_{n}$ be the eigenfunctions of H. Show that $\langle f_{n},f_{m}\rangle=1$ if n=m, and equal to 0 otherwise. Do this by using the creation\annihaltor operators: $a_{+}$ , $a_{-}$ we …
user108605
1
vote
1 answer

Calculate spin wave function given probabilities of its alignment along 2 axes

Problem: An $e^{-}$ exists in such a state that the probability of its spin aligning across the $x_{(+)}$ axis is $P_{x+}=1/2$ and across the $y_{(+)}$ axis is $P_{y+}=1/2$ as well. What is the spin wave function of the electron ? Solution: Let…
stathisk
  • 121
1
vote
1 answer

Complex extension of the Stone - von Neumann Theorem?

I have two questions related to an extension of the Stone - von Neumann Theorem: (1) Are there unitary groups with uncountably many elements indexed over the complex plane? (2) Can the Stone - von Neumann Theorem be formulated over $\mathbb {C}$…
Joe Doe
  • 19
  • 2
1
vote
1 answer

How is the commutator of two vectors of operators in Quantum Mechanics defined?

I'm in a quantum mechanics class, and I have some questions about how the commutator in some cases; what I know is that: if $\hat{A}$ and $\hat{B}$ are operators, their commutator is defined as $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$. (A…
Spherk
  • 157
  • 8
1
2
3 4 5 6 7