Matrix elements are the components, or entries, of a matrix, typically considered in a certain basis.
Matrix elements are the components, or entries, of a matrix, typically considered in a certain basis.
Questions tagged [matrix-elements]
258 questions
18
votes
2 answers
When Eigenfunctions/Wavefunctions are real?
When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real?
What happens in 1D case like the finite quantum well symmetric with respect to the origin…
Lorniper
- 612
18
votes
4 answers
How do one show that the Pauli Matrices together with the Unit matrix form a basis in the space of complex 2 x 2 matrices?
In other words, show that a complex 2 x 2 Matrix can in a unique way be written as
$$
M = \lambda _ 0 I+\lambda _1 \sigma _ x + \lambda _2 \sigma _y + \lambda _ 3 \sigma_z
$$
If$$M = \Big(\begin{matrix}
m_{11} & m_{12} \\
m_{21} & m_{22}
…
Turbotanten
- 1,175
15
votes
1 answer
Is it always possible to express an operator in terms of creation/annihilation operators?
I'm referring to "Path integral approach to birth-death processes on a lattice", L. Peliti, J. Physique 46, 1469-1483 (1985), available at: http://people.na.infn.it/~peliti/path.pdf
The article is about a reformulation of the master equation for a…
zakk
- 1,456
13
votes
3 answers
Definition of an operator in quantum mechanics
In J.J. Sakurai's Modern Quantum Mechanics, the same operator $X$ acts on both, elements of the ket space and the bra space to produce elements of the ket and bra space, respectively. Mathematically, an operator is simply a map between two spaces. …
D12ac
- 141
11
votes
2 answers
Matrix elements of the free particle Hamiltonian
The Hamiltonian of a free particle is $\hat H = \frac{\hat p^2}{2m}$, in position representation
$$ \hat H = -\frac{\hbar^2}{2m} \Delta \;. $$
Now consider two wave functions $\psi_1(x)$ and $\psi_2(x)$ which are smooth enough (say $C^\infty$), have…
Noiralef
- 7,463
10
votes
4 answers
Intuitive argument for symmetry of Lorentz boosts
The Lorentz boosts are represented by symmetric $4\times4$ matrices. Though the most general Lorentz transformations has no obvious symmetry property, can the symmetry (under transpose) of the Lorentz boost matrices be understood intuitively? Like,…
Faber Bosch
- 864
8
votes
4 answers
Matrix element of powers of position operator for quantum harmonic oscillator
A similar question has been asked here before, but that did not contain the particular solution I am after and is now closed. I was wondering if there is a compact analytical formula for matrix elements of the form
$$
\langle m|\hat{x}^k|n…
Quantum
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8
votes
1 answer
Calculation of Matrix Element in "Old-Fashioned Perturbation Theory"
I would like to better under the manipulations/formalism applied in order to evaluate the following matrix element from Schwartz "Quantum Field Theory and the Standard Model" (Eq. 4.16)
$$\quad V _ { n i } ^ { ( R ) } = \left\langle \psi _ { e } ^…
EdRich
- 311
8
votes
2 answers
Matrix Representations of Quantum States and Hamiltonians
I am a high school student trying to teach himself quantum mechanics just for fun, and I am a bit confused. As a fun test of my programming/quantum mechanics skill, I decided to create a computer program to model an atom.
Initially, I learned about…
Shivam Sarodia
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- 34
7
votes
4 answers
Table of matrix elements of powers of $r$ for radial functions in $H$ atom
Im looking for some references here. I hope it is the right place to ask.
I need to find a table of (or a formula from which to extrapolate) the matrix elements of the radial functions of the hydrogen atom evaluated in powers of $r$ for both…
user24273
- 141
6
votes
1 answer
Why doesn't the Weinberg-Witten theorem forbid collinear photons?
The Weinberg-Witten theorem tells us that any theory that has an effective graviton, i.e. a massless helicity-2 particle as a state in the free-particle Fock space, cannot have a gauge-invariant and Lorentz-covariant stress-energy tensor that gives…
Ahron Maline
- 260
6
votes
1 answer
Photon-Gluon annihilation in QCD
I am starting to learn about QCD, and I wanted to calculate the squared matrix elements for photon-gluon annihilation into a quark and an anti-quark. However, I am having trouble writing down the correct matrix elements at tree level and computing…
Joel
- 305
6
votes
3 answers
truncation before matrix exponential: how to do it right?
I'm trying to compute (numerically) the matrices of some simple quantum optical operations, which in principle are unitary.
However, in my case they are unitary in an infinite-dimensional space, so I have to truncate them. The result is not…
Ziofil
- 258
6
votes
4 answers
What type of mathematical object is the "Pauli vector"?
The three pauli matrices $\sigma_x$, $\sigma_y$, $\sigma_z$ are sometimes combined in the "Pauli vector", usually denoted $\boldsymbol{\sigma} = \sigma_{x} \boldsymbol{e_x} + \sigma_y \boldsymbol{e_y} + \sigma_z \boldsymbol{e_z}$, the intuitive…
Ignacio
- 1,330
6
votes
2 answers
How are matrices used to represent quantities, and what is the meaning of a matrix?
So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy of an object, are represented in a matrix, and…
Addem
- 1,249