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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7
votes
3answers
357 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 $P_{...
5
votes
2answers
574 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 ...
1
vote
3answers
76 views

Explanation of this integral

$$\int_{-\infty}^{\infty} e^{-\frac{i}{\hbar}(p-p')x} dx = 2\pi\hbar\delta(p-p')$$ I don't quite understand how this integration leads to the right hand side. Any explanation is appreciated.
4
votes
1answer
45 views

Tensor product in dual-space

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
15
votes
4answers
534 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
3
votes
5answers
482 views

How to prove $e^{A \oplus B} = e^A \otimes e^B$ where $A$ and $B$ are matrices? (Kronecker operations)

How to prove that $e^{A \oplus B} = $$e^A \otimes e^B$? Here $A$ and $B$ are $n\times n$ and $m \times m$ matrices, $\otimes$ is the Kronecker product and $\oplus$ is the Kronecker sum: $$ A \oplus B =...
3
votes
2answers
543 views

Expected Values of Operators in Quantum Mechanics

I've recently started an introductory course in Quantum Mechanics and I'm having some trouble understanding what the expectation of an operator is. I understand how we get the formula for the ...
1
vote
0answers
417 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output. [on hold]

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
4
votes
1answer
170 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
2
votes
0answers
117 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 ...
2
votes
1answer
101 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: $$T_\...
1
vote
1answer
36 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
1
vote
2answers
97 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
0
votes
1answer
41 views

Determinant Expression for Grover operator

I want find the characteristic polynomial for the grover quantum operator $U^{N\times N}$ $$\begin{align*} U=(2|D\rangle\langle D| -I_N)(2|M^{\perp}\rangle\langle M^{\perp}| -I) \end{align*}$$ ...