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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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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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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$\sup_{0 \leq M \leq I_{\mathcal{H}_1}} \frac{\operatorname{Tr}[\rho M]}{\operatorname{Tr}[\sigma M]}$ 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
$$D(\rho\parallel\sigma) := \sup_{0 \leq M ...
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Show that a state $\rho=\sum_i p_i|e_i\rangle\!\langle e_i|$ has purifications of the form $\sum_i s_i |e_i\rangle\otimes|f_i\rangle$
Let $ρ_A = \sum_{i=1}^r p_i|e_i⟩⟨e_i|$, where $p_i$ are the nonzero eigenvalues
of $ρ_A$ and $|e_i⟩$ corresponding orthonormal eigenvectors. If some eigenvalue appears more
than once then this ...
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About the idea that, the "Normalization" of a "scattering wavefunction", being insensitive to the functions form near to the scattering centre.
I thought I saw ..., well, not a pussy cat, but a comment in Dirac$^1$, going on about the "Normalization" of a scattering wave function being insensitive to the form of such a wavefunction ...
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A problem with analysing the 'Delta Function Normalization', of an 'Irregular Coulomb Wave Function'.
Edit May 24th 2022
Please note, I no longer think that
\begin{equation*}
\int _0^\infty G_L(\eta,~k^\prime r) G_L(\eta,~kr) dr = N_{L,G} \delta(k-k^\prime)
\end{equation*}
for some real number $N_{L,G}...
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1
answer
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What is the essential difference between classical and quantum information geometry?
This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory.
I have a ...
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Two Dirac delta functions in an integral?
For context, this is from a quantum mechanics lecture in which we were considering continuous eigenvalues of the position operator.
Starting with the position eigenvalue equation,
$$\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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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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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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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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Deriving Rodrigues Formula and Generating function of Hermite Polynomial from $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$
There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, ...
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Does every operator have a hermitian adjoint?
If we think of operators as matrices, every matrix can be transposed and its elements can be complex-conjugated. But the identification of the hermitian adjoint with the transpose conjugate comes from ...
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Show that a set of projectors summing to the identity implies mutually orthogonal projectors
The general setting is the study of positive operator measures in quantum mechanics,
instead of the projector operator measures.
Going from the PVM to the POVM is just saying that our bunch of ...
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Is the Unitary Group of a Hilbert Space a Lie group?
Let $H$ be an infinite-dimensional complex Hilbert space. Then the set of unitary operators on $H$ forms a group, known as the unitary group or Hilbert group. My question is, is this group a Lie ...
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Possible significant error in proof of the spectral theorem, Brian C Hall, Quantum Theory for Mathematicians
Note/Edit: Read the below paragraphs for context. I think I found a counterexample, though I don't have the energy to work through it now. Suppose $\mathcal{H} = L^2([0,1],m) \oplus L^2([0,1],m)$ ...
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How does one define the complex distribution $1/z$?
I have read the following formula in a quantum mechanics book, supposedly attributed to Dirac
$$ \lim_{y\,\searrow\, 0} \frac 1 {x+iy} = \operatorname{p.v.} \left(\frac 1 x\right) - i \pi\delta (x)$$
...
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Does the split-step operator method work for a PDE in cylindrical coordinates?
I am trying to numerically solve the 3D Schrodinger equation with an extra non-linear term (the Gross-Pitaevskii equation) using the split-step operator method. This is a follow up to a previous ...
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Geometric quantization: not understanding the curvature form and Weil's theorem
I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
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Expected Values of Operators in Quantum Mechanics
I've recently started an introductory course in Quantum Mechanics and I'm having some trouble understanding what the expectation of an operator is. I understand how we get the formula for the ...
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2
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Proving that the Laguerre polynomials do indeed solve the differential equation
I am trying to show that the Laguerre differential equation, given in my homework problem as
$xL''_n(x)+(1−x)L'_n(x)+ nL_n(x) = 0$,
is indeed solved by the Laguerre polynomials in their closed sum ...
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Dirac Delta function identities
I'm using the definition of the one dimensional dirac-delta function, $\delta(x)$, being, $$\int_{-\infty}^{+\infty}\delta(x)f(x)dx = f(0) \tag1$$ and I'm doing a question which asks me to (via ...
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When does this integral vanish, which appears in the derivation of the quantum virial theorem?
In the derivation of the quantum virial theorem by Slater (appendix of this article), the following term $A$ appears, which is said to vanish upon integration ($\bar \psi$ is the complex conjugate of $...
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Understanding the Expression $tr\Big(\rho(X\otimes I)\Big)=\sum_{a,b,a',b'} \rho_{ab,a'b'}X_{a,a'}\delta_{b,b'}$
How do I make sense of
$$
tr\Big(\rho(X\otimes I)\Big)=\sum_{a,b,a',b'} \rho_{ab,a'b'}X_{a,a'}\delta_{b,b'}
$$
If this does not involve any tensor products it would be a simple elementwise matrix ...
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Given product and convolution of pair of functions can you find original pair?
Suppose you have two functions $f,g: \mathbb{R} \rightarrow \mathbb{C}$ and you have their product and convolution
$$ h_1(t) = fg$$
$$ h_2(t) = \int_{-\infty}^{\infty} f(t-\tau)g(\tau) d\tau $$
Is it ...
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Why does $\Omega_{ij} = \overline{\Omega}_{ji}$ only imply $\Omega$ is Hermitian in the finite dimensional case?
1 Question for Bounty
In the context of an infinite-dimensional vector space, below I present a supposed "proof" – designated by $(*)$ – that
$$
\Omega_{ij} = \overline{\Omega}_{ji}
$$
...
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1
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Evaluating Gibbs state in the second quantized formalism
First, let us fix our notation. If $A:\mathcal H \to \mathcal H$ is a linear operator on a single-particle Hilbert space $\mathcal H$, we can lift $A$ on the Fock space $\mathfrak F_{s/a}(\mathcal H)$ ...
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Squaring an operator
There is an excercise of squaring an operator in my book of quantun mechanics.
The operator is $$\hat{A}=\frac{\mathrm{d}}{\mathrm{d}x}+x$$
And I should compute $\hat{A}^2$. He gives me a result
$$\...
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votes
0
answers
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Eigenvalue of an Euler product type operator?
Background
Let us have the following orthonormal basis such that:
$$ \langle m | n \rangle = \delta_{mn}$$
Consider the following operators defined as:
$$ \hat 1 = | 1 \rangle \langle 1 | + | 2 \...
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Quantum Mechanics: position and the separability of Hilbert space?
I would be pleased if someone could point out to me where I go wrong in the following sequence of statements:
One model of quantum mechanics identifies states of a particle with normalized vectors (...
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1
answer
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How can we show that a map is a completely positive map?
I am doing a homework problem where I have to find whether the map
$$ \rho ~\rightarrow~ {\rm tr}(\rho) I - \rho $$ is completely positive.
If the map is not completely positive, a counter-example ...
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