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:
$$
A \oplus B =...
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Regarding Ladder Operators and Quantum Harmonic Oscillators
When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator:
Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
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Tensor product in dual-space
I am a bid confused regarding the notation for tensor products when going into dual-space
If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
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A question on use of square integrable functions
I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour).
As far as I understand it, a square-integrable function $f(x)$ satisfies the condition $$\...
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How do you work out, the 'Delta Function Normalization', of a 'Regular Coulomb Wave Function'?
A one dimensional momentum eigenfunction $ e^{ikx}$, can be normalized in terms of the 'Dirac Delta Function', because we can write that, for some value of $C$
\begin{equation*}
\int_{-\infty}^\infty ...
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Mazur-Ulam-like theorem for complex Hilbert spaces
The Mazur-Ulam theorem doesn't hold for complex Hilbert spaces because antiunitary operators are origin preserving surjective isometries, but they aren't linear. Is it true, that every $f:H\to H$ ...
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$\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{\operatorname{Tr}[\rho M]}{\operatorname{Tr}[\sigma M]}$ satisfies Data Processing Inequality
$\newcommand{\Tr}{\operatorname{Tr}}$Let $\mathcal{H}_1, \mathcal{H}_2$ be a Hilbert spaces and $\rho, \sigma$ be density matrices on $\mathcal{H}_1$. Define
$$D(\rho\parallel\sigma) := \sup_{0 \leq M ...
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About the idea that, the "Normalization" of a "scattering wavefunction", being insensitive to the functions form near to the scattering centre.
I thought I saw ..., well, not a pussy cat, but a comment in Dirac$^1$, going on about the "Normalization" of a scattering wave function being insensitive to the form of such a wavefunction ...
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Show that a state $\rho=\sum_i p_i|e_i\rangle\!\langle e_i|$ has purifications of the form $\sum_i s_i |e_i\rangle\otimes|f_i\rangle$
Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues
of $ρ_A$ and $|e_i⟩$ corresponding orthonormal eigenvectors. If some eigenvalue appears more
than once then this ...
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A problem with analysing the 'Delta Function Normalization', of an 'Irregular Coulomb Wave Function'.
Edit May 24th 2022
Please note, I no longer think that
\begin{equation*}
\int _0^\infty G_L(\eta,~k^\prime r) G_L(\eta,~kr) dr = N_{L,G} \delta(k-k^\prime)
\end{equation*}
for some real number $N_{L,G}...
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What is the essential difference between classical and quantum information geometry?
This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory.
I have a ...
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Two Dirac delta functions in an integral?
For context, this is from a quantum mechanics lecture in which we were considering continuous eigenvalues of the position operator.
Starting with the position eigenvalue equation,
$$\hat{x}\,\phi(x_m, ...
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Prerequisite for Takhtajan's "Quantum Mechanics for Mathematicians"
I want to know the math that is required to read Quantum Mechanics for Mathematicians by Takhtajan.
From the book preview on Google, I gather that algebra, topology, (differential) geometry and ...
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Why do physicists get away with thinking of the Dirac Delta functional as a function?
For instance they use it for finding solutions to things like Poisson's Equation, i.e. the method of Green's functions.
Moreover in Quantum Mechanics, it's common practise to think of the delta ...
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How do outer products differ from tensor products?
From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
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Baker Campbell Hausdorff formula for unbounded operators
Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have
$$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$
which for $[X,Y]$ being central reduces to
$$e^Xe^Y=\exp(X+Y+\frac12[X,Y])....
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Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?
Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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Probability and Quantum mechanics
I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism.
To wit, we usually say that an observable ...
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