Questions tagged [normalization]
The normalization tag has no usage guidance.
244
questions
1
vote
1answer
65 views
About constants in fermionic path integral in Peskin and Schroeder
I am confused by fermionic path integral used in Peskin and Schroeder. Equation (9.69) gives
$$\Big(\prod_n\int d\bar{\theta}_nd\theta_n\Big)e^{-\bar{\theta}_iM_{ij}\theta_j}=\det M\tag{9.69}$$
But ...
1
vote
0answers
52 views
How to find probability of finding a particle outside the region in which it is confined?
Problem:
Consider a one-dimensional particle of mass $m$ which is confined within the region $0 \le x \le a$ by a potential $V(x)$ and whose wave function is $\Psi(x, t) = \sin(\pi x/a)\exp(−i\omega t)...
0
votes
0answers
59 views
Explanation of an worked example from N.Zettili Quantum Mechanics
In the book "Quantum Mechanics Concepts and Application" 2nd edition by N.Zettili the worked problem 1.11.
Here I had to find the Fourier transform of the function,
$$\phi(k) =
\begin{cases}...
2
votes
1answer
82 views
How do I prove that an electron beam has a plane wave function?
I have been told that an electron beam has a wave function equivalent to a plane wave $\psi(x) = Ae^{ikx}$, however I would like to know why? Also, if an electron beam can be shown to have a wave ...
-1
votes
4answers
88 views
Paradox regarding non-normalizable wavefunctions
Suppose that we have a quantum particle with defined momentum $p=\hbar k$, this means that the particle is in an eigenstates of the momentum operator $|k \rangle$. We can now ask: what is the ...
0
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0answers
26 views
Canonical Partition function of constant gravitational potential: what is the normalization?
I'm imagining a system of particles subject to a constant gravitational field, say $V(x)=g x$. The Hamiltonian for one particle will be
$$H=\frac{1}{2}m v^2 + gx ,$$
where the first term is the ...
0
votes
0answers
30 views
Normalization of functions in function space
It is a well know question, be the function space $L_{2}(-\infty,\infty)$, evidently the eigenfunctions of $$i\frac{d}{dx}, X$$ are $$f = Ae^{-i \lambda x}, g=B \delta (x-\lambda)$$ respectively.
...
0
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0answers
21 views
Normalization for the Fourier series of a Gaussian Electric field
I have a function $f(x) = \exp(-\frac{x^2}{\sigma^2})$ on a box of size $L$. I have the given relation between the size of the box and the width of the Gaussian $\sigma = \sqrt{\frac{2}{\pi}}L$. I use ...
0
votes
2answers
78 views
Normalising a wave-function
So I have a small confusion when normalising an infinite well wave-function. The wave-function for my problem is $$Ψ(x) = Ae^{i(kx-wt)}+Be^{-i(kx-wt)}+Ce^{i(kx-wt)}+De^{-i(kx-wt)}.\tag{1}$$
Applying ...
0
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0answers
29 views
How is the nuclear wavefunction norm defined in normal coordinates?
Lets assume we have a diatomic molecule with a total of six Cartesian coordinates. Lets also assume that the BO-Approximation is valid and we can write the ground state wavefunction like this,
$$
\...
2
votes
2answers
502 views
Why do we refer the cross section ratios to muons?
For electron-positron interactions, we have different cross sections, depending on the available reaction energy. To get an overview how many particles of a certain type have been created, we can ...
1
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0answers
68 views
Is probability conserved in Dirac's original formulation of relativistic QM?
In non-relativistic quantum mechanics, the normal condition for position eigenstates is
$$\langle y|x\rangle=\delta(y-x).$$
However, this condition is not Lorentz-invariant. I have never seen a ...
4
votes
2answers
122 views
How to Normalize a Wave Function?
To talk about this topic let's use a concrete example:
Suppose I have a one-dimensional system subjected to a linear potential, such as the hamiltonian of the system is:
$$H=\frac{\hat{p}^2}{2m}-F\hat{...
2
votes
0answers
102 views
How to justify unit cell normalization of Bloch functions?
As is widely known, non-square-integrable wavefunctions don't belong to the Hilbert space, and therefore cannot represent physical states.
This is the case for e.g. oscillating wavefunctions and ...
1
vote
1answer
97 views
Two non-interacting particles in a 1D box
I need to find the wave function for two non-interacting particles of mass m_1 and m_2 in 1D infinite box (well) of length $L$, where the positions of the particle is given by $x_i$ ($i$ being 1,2).
...
0
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1answer
83 views
Quantum Mechanics (Griffiths); Am I Missing A Subtle Argument In A Proof?
I'm working through a proof in Griffith's Quantum Mechanics book (Chapter 1.4 - Normalization) and feel like a subtle detail is being omitted. If anyone can supply clarity that would help.
We have $\...
0
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0answers
51 views
Wave functions without normalization
I'm working on a problem where we find the probability of finding a particle between $a/4$, and $3a/4$. The particle is confined between $0$ and $a$.
Normally I would attack this by normalising the ...
-1
votes
1answer
55 views
How do I normalize the complex interference of two states?
Let me define two complex-valued functions as follows:
$$
\psi_1(r_1,\theta_1)=r_1e^{i\theta_1}\\
\psi_2(r_2,\theta_2)=r_2e^{i\theta_2}
$$
By the Born rule I can calculate the probability density, a ...
-2
votes
2answers
66 views
How can you calculate the normalisation factor? [closed]
when given a particle of mass $m$ in a one dimensional square potential well from $x=0-L$, How can one calculate the value of the normalisation factor $|φ\rangle = |A|(\,|1\rangle + |2\rangle\,)$ ...
0
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2answers
72 views
Schrodinger Equation of a Hydrogen Atom in a Rydberg State
I am not a mathematician, so I really appreciate it if someone could explain it in a simple way. In Rydberg Atoms book by Thomas F. Gallagher, the Schroedinger equation for the H atom in atomic units ...
4
votes
1answer
117 views
Why do different quantum field theory books have different conventions regarding the normalization? [closed]
Why are there several different versions of the scalar field solution, with a different coefficient in front of the exponential in the solution? Why don't all the authors use the same convention? I ...
0
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0answers
26 views
How does one normalize an $n$-particle wavefunction composed of non-orthogonal one or two-particle basis functions?
I am interested in constructing a trial wavefunction for Diffusion Monte Carlo, of the Valence Bond form. If the multi-fermion wf is constructed from two-particle basis elements which are non-...
1
vote
2answers
207 views
Why do we normalize the radial and the angular parts of a spherical wavefunction separately?
I'm trying to revise the Quantum Mechanical model of the Hydrogen atom, and I understand all the methods involved, including separating the wavefunction into its radial and angular parts, solving all ...
2
votes
2answers
89 views
Infinite correlation functions in free field theory
In a free scalar field theory, Wick's theorem guarantees that $\langle \hat\phi(x)\rangle = 0$ and $\langle \hat\phi(x)^2\rangle = \infty$. Given that $\hat \phi(x)$ creates a particle at $x$, these ...
0
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0answers
27 views
How to normalize a deconvolution of a signal?
What I have:
A recorded signal, in dimensions voltage (or any kind of intensity) over time
A response function of the system, responding to a nearly gaussian input signal
What I try to achieve:
I ...
0
votes
0answers
35 views
Normalization for Clebsch-Gordan Decomposition in Symmetrized Spinor Representation
In various references, including Rovelli's "Quantum Gravity," a method is presented for calculating the Clebsch-Gordan decomposition of two spin states. Suppose we have two spins with $j_{1}$...
1
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1answer
65 views
Is this integral always equal to 1?
This is my Hamiltonian. $\psi_{\alpha}$ is a bosonic field.
$$H_{\alpha}=\int \mathrm{d} \mathbf{r} \psi_{\alpha}^{\dagger}(\mathbf{r})\left(-\frac{\nabla^{2}}{2 m}\right) \psi_{\alpha}(\mathbf{r})+\...
5
votes
1answer
155 views
Weinberg QFT 1 Normalization one 1 particle states p. 66
I encounter a question regarding the derivation of the normalization of 1 Particle states in Weinbergs book (Formula 2.5.14).
Similar questions were asked in A question on page 65 of Weinberg's ...
0
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1answer
134 views
Normalization for a free Dirac plane wave
I've recently come about the free plane wave solutions to the Dirac equation, and i'm having a hard time proving that the normalization factor $n$ is
$$n=\frac{1}{\sqrt{2m(m+\omega)}}$$
Where $\omega$ ...
3
votes
1answer
76 views
Normalizing symmetric wavefunctions
Wikipedia and other sources say that the normalized symmetric ket for $N$ particles with quantum numbers $n_1, n_2, ...,n_N$ is
$$|n_1n_2...n_N;S\rangle=\sqrt{\frac{\Pi_km_k!}{N!}}\sum_P|n_{P(1)}\...
-1
votes
1answer
163 views
Normalisation of the following wavefunction: $\psi(\theta,\phi)=\cos(\theta)$ [closed]
Normalisation of the following wavefunction:
$\psi(\theta,\phi)=\cos(\theta)$
So I thought about setting the following
$N\int \cos(\theta)\cos^*(\theta) d\theta=1$
But then maybe I thought I was ...
0
votes
3answers
117 views
Physical Significance of non-normalized state
What does the coefficient physically mean for an operator that isn't an observable. For an observable the coefficient is the eigenvalue and is the value that will be measured, but for operators that ...
1
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0answers
44 views
Normalization of Generators of $SU(N)$
I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the ...
0
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1answer
97 views
How to write the time-dependent Schrödinger equation from generic functions?
Given the initial state: $$\Psi(x,t=0)=c_1 \psi_1(x)+c_2\psi_2(x)+c_yy(x)$$ where $\psi_1$ and $\psi_2$ are eigenstates of $\hat{H}$ and $y(x)$ is a normalizable function but is not eigenstate of $\...
-1
votes
1answer
53 views
Why doesn't integration of the wavefunction squared give 1 for rigid rotors?
This source states that the wavefunction for rigid rotors for a constant radius $R$ is:
$$\psi(\theta,\phi)=\Theta(\theta)\cdot \Phi(\phi)$$
Integrating $\psi(\theta,\phi)^2$ over $d\theta$ and $d\...
1
vote
1answer
224 views
Normalization of a Wave-function in spherical co-ordinates [closed]
So I have been provided with the following wave-function
$ψ(x, y, z) = N(x + y + z)e^\frac{
−r^
2}{α^2}$
I am trying to convert it to spherical co-ordinates and to find the Normalization constant $N$...
0
votes
2answers
97 views
Normalization in perturbation theory
When we have a system with hamiltonian $H = H_{0} + V$, we can expand the ground state wavefunction $\Psi_{0}$ using the wavefunction of the non-interacting system $\phi_{0}$, that is an eigenfunction ...
3
votes
2answers
476 views
Can momentum never be zero in quantum mechanics?
I have seen Zetilli's QM book deals with $E>V$ and $E< V$ (tunnelling) in case of the potential wells deliberately avoiding the E=V case,So I thought maybe something is intriguing about this and ...
2
votes
1answer
68 views
Green's function for the screened Poisson equation
Assuming we are given a Lagrangian
\begin{equation}
\mathcal{L}(\phi(r),\partial^i\phi(r)) = \frac{1}{2} \partial_i\phi \partial^i\phi + \frac{m^2}{2} \phi^2 + \lambda \phi,
\end{equation}
the ...
1
vote
0answers
40 views
Normalization constant of the collision wave function [closed]
The problem
The quantum system under consideration is the 1-dimensional system with the step potential
\begin{equation}
\mathcal{V} =
\begin{cases}
0 \quad & x<0\\
U_0 \quad & 0&...
1
vote
1answer
191 views
Confusion over QM Free Particle Propagator - Shankar 5.1 [duplicate]
I am working through Shankar's Principles of Quantum Mechanics and I am confused about his derivation for the propagator of a free particle from the Schrodinger Equation (Section 5.1).
He states that ...
1
vote
1answer
88 views
Why does $\phi=\phi^*$ imposed on complex scalar field Lagrangian miss out $1/2$ factors?
If we require the reality condition $\phi=\phi^*$ on the Lagrangian for a complex scalar field is $$\mathcal{L}=(\partial^\mu\phi^*)(\partial_\mu\phi)-m^2(\phi^*\phi),$$ two degrees of freedom $\phi$ ...
2
votes
2answers
275 views
The use of $a^\dagger(\mathbf{k}) = -i \int d^3x e^{ikx}\stackrel{\leftrightarrow}{\partial}_0 \phi(x)$ in the derivation of the LSZ-formula
I noticed that in Srednicki's derivation of the LSZ-formula the expression (chapter 5) for the creation (and also later for the annihilation) operator by the field operator:
$$a^\dagger(\mathbf{k}) = -...
1
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0answers
93 views
Basic Question about Quantum Mechanics
I have a basic question about quantum mechanics. Let's say we consider one particle in one space dimension. In my understanding, in quantum mechanics, the location is given by a distribution function $...
1
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1answer
43 views
How can I find the normaliztion constant of probabilty amplitude for small space-time path?
For free particle with $V=0$ case we get
$$<x_n,t_n;x_{n-1},t_{n-1}>=\frac{1}{w(\Delta t)}\exp \left [\frac{im(x_n-x_{n-1})^2}{2\hbar\Delta t}\right ] \tag{6.42}$$
given in eqn. 6.42 of ...
-1
votes
1answer
83 views
How to normalize this wave function? [closed]
My wave function is
$$
\Psi = A e^\left({-\frac{\left|x\right|}{2a}- \frac{\left|y\right|}{2b} -\frac{\left|z\right|}{2c}}\right)dx
$$
and I need to normalize it. I tried to take an integral of it and ...
2
votes
1answer
102 views
Why isn't it a problem that the first term in the perturbative scattering series yields infinity (or one)?
In QFT we usually want to calculate objects of the form $\langle f|\hat S|i\rangle$ which yields the probability amplitude for the process $i \to f$. We can expand the scattering operator $\hat S$ in ...
0
votes
0answers
81 views
Verification of the normalization of a wave function
A particle in an l-dimensional box is described by the wavefunction
Probability density = 1 to be normalized but I can't proceed from the last step
$$P= 1 - (1/L) (\sin 2n\pi\cos 2n\pi)$$ to equal to ...
0
votes
1answer
52 views
What happened to the factor of $\pi$ in this question?
$\\
$
I was going through the answer to this problem, when I noticed that a factor of $\pi$ in the denominator disappeared and a factor of 4 appeared in the numerator when the author started ...
0
votes
1answer
48 views
Property of the time evolution operator preserving the norm of the wavefunction
Since the time evolution unitary operator preserves norm, if applied to any system say electron whizzing around its orbitals, no matter what time we consider would it always have the same probability ...
