Questions tagged [quantum-mechanics]
For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.
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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_{...
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Fourier Representation of Dirac's Delta Function
This question is related to this other question on Phys.SE.
In quantum mechanics is often useful to use the following statement:
$$\int_{-\infty}^\infty dx\, e^{ikx} = 2\pi \delta(k)$$
where $\delta(k)...
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Quantum mechanical books for mathematicians
I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential ...
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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 ...
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Quantum mechanics for mathematicians
I'm looking for books about quantum mechanics (or related fields) that are written for mathematicians or are more mathematically inclined.
Of course, the field is very big so I'm in particular ...
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Clean proof of Baker-Campbell-Hausdorff Formula
I am thinking of the cleanest way to prove the BCH formula and I have come up with this.
First, work out $e^{\lambda A}Be^{-\lambda A}$ by expanding the exponentials (sums go from $0$ to $\infty$):
$$\...
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If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?
I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
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Books on Perturbation Methods
I am having problems finding descent books on perturbation methods. I am looking for a book which covers; asymptotic expansions, matched Asymptotic expansions, Laplace's Method, Method of steepest ...
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Nielsen & Chuang, exercise 2.73 — Density matrix proving the minimum ensemble
I've been trying to solve exercise 2.73 (p.g 105) in Nielsen Chuang, and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'...
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Explanation of this integral
$$
\int_{-\infty}^{\infty}
\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}\left(p - p'\,\right)x} \,\,\,\mathrm{d}x =
2\pi\hbar\,\delta\left(p - p'\right)
$$
I don't quite understand how this integration leads ...
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Why do odd dimensions and even dimensions behave differently?
It is well known that odd and even dimensions work differently.
Waves propagation in odd dimensions is unlike propagation in even dimensions.
A parity operator is a rotation in even dimensions, but ...
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Mathematics needed in the study of Quantum Physics
As a 12th grade student , I'm currently acquainted with single variable calculus, algebra, and geometry, obviously on a high school level. I tried taking a Quantum Physics course on coursera.com, but ...
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How to prove $e^{A \oplus B} = e^A \otimes e^B$ where $A$ and $B$ are matrices? (Kronecker operations)
How to prove that $e^{A \oplus B} = $$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 =...
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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 ...
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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 ...
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