Questions tagged [normalization]
The normalization tag has no usage guidance.
215
questions
0
votes
0answers
49 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
36 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
$$|n_1n_2...n_N;S\rangle=\sqrt{\frac{\Pi_km_k!}{N!}}\sum_P|n_{P(1)}\...
-1
votes
1answer
78 views
Normalisation of the following wavefunction: $\psi(\theta,\phi)=\cos(\theta)$ [closed]
Normalisation of the following wavefunction:
$\psi(\theta,\phi)=\cos(\theta)$
So I thought about setting the following
$N\int \cos(\theta)\cos^*(\theta) d\theta=1$
But then maybe I thought I was ...
0
votes
3answers
88 views
Physical Significance of non-normalized state
What does the coefficient physically mean for an operator that isn't an observable. For an observable the coefficient is the eigenvalue and is the value that will be measured, but for operators that ...
1
vote
0answers
29 views
Normalization of Generators of $SU(N)$
I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the ...
0
votes
1answer
33 views
Constructing solution to the time-dependent Schrödinger's equation
Given the initial state: $$\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)$$
Integrating $\psi(\theta,\phi)^2$ over $d\theta$ and $d\...
1
vote
1answer
88 views
Normalization of a Wave-function in spherical co-ordinates [closed]
So I have been provided with the following wave-function
$Ļ(x, y, z) = N(x + y + z)e^\frac{
ār^
2}{Ī±^2}$
I am trying to convert it to spherical co-ordinates and to find the Normalization constant $N$...
0
votes
2answers
53 views
Normalization in perturbation theory
When we have a system with hamiltonian $H = H_{0} + V$, we can expand the ground state wavefunction $\Psi_{0}$ using the wavefunction of the non-interacting system $\phi_{0}$, that is an eigenfunction ...
3
votes
2answers
371 views
Can momentum never be zero in quantum mechanics?
I have seen Zetilli's QM book deals with $E>V$ and $E< V$ (tunnelling) in case of the potential wells deliberately avoiding the E=V case,So I thought maybe something is intriguing about this and ...
2
votes
1answer
55 views
Green's function for the screened Poisson equation
Assuming we are given a Lagrangian
\begin{equation}
\mathcal{L}(\phi(r),\partial^i\phi(r)) = \frac{1}{2} \partial_i\phi \partial^i\phi + \frac{m^2}{2} \phi^2 + \lambda \phi,
\end{equation}
the ...
1
vote
0answers
38 views
Normalization constant of the collision wave function [closed]
The problem
The quantum system under consideration is the 1-dimensional system with the step potential
\begin{equation}
\mathcal{V} =
\begin{cases}
0 \quad & x<0\\
U_0 \quad & 0&...
1
vote
1answer
123 views
Confusion over QM Free Particle Propagator - Shankar 5.1 [duplicate]
I am working through Shankar's Principles of Quantum Mechanics and I am confused about his derivation for the propagator of a free particle from the Schrodinger Equation (Section 5.1).
He states that ...
1
vote
1answer
64 views
Why does $\phi=\phi^*$ imposed on complex scalar field Lagrangian miss out $1/2$ factors?
If we require the reality condition $\phi=\phi^*$ on the Lagrangian for a complex scalar field is $$\mathcal{L}=(\partial^\mu\phi^*)(\partial_\mu\phi)-m^2(\phi^*\phi),$$ two degrees of freedom $\phi$ ...
2
votes
2answers
175 views
The use of $a^\dagger(\mathbf{k}) = -i \int d^3x e^{ikx}\stackrel{\leftrightarrow}{\partial}_0 \phi(x)$ in the derivation of the LSZ-formula
I noticed that in Srednicki's derivation of the LSZ-formula the expression (chapter 5) for the creation (and also later for the annihilation) operator by the field operator:
$$a^\dagger(\mathbf{k}) =...
1
vote
0answers
92 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
vote
1answer
41 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
77 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
97 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
34 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
51 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
34 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 ...
-1
votes
2answers
93 views
Probability in quantum physics of a wave function
I have this time-dependent wave function from solving a 3-component Schrodinger's equation, $$\psi(t)=-\frac{2}{9}(-2, 1, 2)^T+\frac{2}{9}e^{3i\omega t}(2,2,1)^T+\frac{1}{9}e^{-3i\omega t}(1, -2, 2)^T$...
1
vote
1answer
62 views
Should a wavefunction in momentum space be normalisable?
Is this a condition that the wavefunction in momentum space should be normalizable? Like we said that a particle has to be between ${-\infty}$ to ${\infty}$. Will the same argument also work for ...
1
vote
2answers
160 views
Quantum Harmonic Oscillator Raising and Lowering operators
The commutator of the operators, $[a,a^\dagger] = 1$ is useful in rewriting the Hamiltonian in a neat way in terms of the creation and annihilation operators.
So my question is, Is there a physical ...
0
votes
1answer
63 views
Solutions that are part of the Hilbert space
Why do we omit solutions that do not converge at $\pm\infty$ from the physical Hilbert space, what is the argument for us being allowed to do so?
-1
votes
2answers
65 views
Proving the preservation for the norm of a wavefunction
How can we prove the preservation for the norm of the wave-function for a specific hamiltonian (say a spin 1/2 particle) for all times?
0
votes
1answer
82 views
Is the normalization of the wave-function preserved due to…?
Is the preservation of the inner product the same thing as the vector length of the wave-function staying constant with it's rotation through some R2 plane (ie it's evolution through time), that is ...
1
vote
2answers
129 views
Solving time-independent Schrodinger equation with $V=0$
I recently came across this section in "Physical chemistry" by Peter Atkins and Julia Paula.
There they discuss the solution of Schrodinger equation for a particle of mass m moving through a single ...
1
vote
2answers
137 views
Interpretation of the propagator
In quantum mechanics, it is clear that $\langle x|y\rangle = 0$ for $x\ne y$, where $|x\rangle$ is the state with the particle at position $x$. (Notice that this $|x\rangle$ is different from the ...
-1
votes
1answer
74 views
Finding normalization constant of a wave function with definite momentum
I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point,
$$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$
This is an expression for ...
2
votes
2answers
145 views
What is the mathematical reasoning behind Schrodinger's equation preserving its normalization, with the evolution of time?
I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of ...
0
votes
2answers
81 views
Some quantum-mechanical questions [closed]
I have recently started studying quantum mechanics, and here are some things that are really confusing me.
Particle in a box: Supposedly, the square of the magnitude of the normalized wave function ...
1
vote
2answers
207 views
Hypercharge normalization for $SU(5)$ GUT
Reading about $SU(5)$ unification, texts says that they use the renormalization factor $\sqrt{3/5}$ for weak hypercharges in order to embed SM into a $SU(5)$ group. This implies a new $U(1)_Y$ ...
3
votes
1answer
167 views
How to derive Eq. (6.21) in Srednicki?
I'm reviewing Srednicki's chapter on path integrals and am having trouble understanding how he arrives at formula 6.21:
$$\left<0|0\right>_{f,h}= \int \mathcal{D}q \,\mathcal{D}p
\, \exp \left[...
0
votes
0answers
18 views
Normalizing Efficiencies with different motor power ratings
I have obtained data of the efficiencies of a few models of electric vehicles. We would like to compare them, but we have noted that the different motor power ratings may have an effect on the ...
0
votes
1answer
64 views
Dirac delta normalization of electromagnetic fields
Usually, to quantize electromagnetic fields we use box normalization and therefore the normalization constant contains the dimensions of Volume V of the box.
But if we perform the Dirac-delta ...
-1
votes
1answer
85 views
Why is there a factor of $\sqrt2$ listed for the neutral pion?
I was reading the Wikipedia page on pions.
At the bottom of the page, there is a listing for the neutral pion ($\frac{\rm u\bar u+d\bar d}{\sqrt{2}}$).
Why are they over the square root of 2?
There ...
0
votes
1answer
89 views
What are exactly “norms” in spin networks? Are there any non-quantum spin networks?
Roger Penrose proposed a series of networks from which, fundamentally, space-time would emerge, called spin networks
(https://en.wikipedia.org/wiki/Spin_network)
In this article, it is said:
...
0
votes
4answers
172 views
If a wavefunction is normalized at $(x,t)$, is it also normalized at $(x\!-\!ct,t)$?
If $\Bigl( \Psi(x,t),\Psi(x,t) \Bigr) =1$, I want to find out if $\Bigl( \Psi(x-ct,t),\Psi(x-ct,t) \Bigr) =1$.
My attempt was
$\Bigl( \Psi(x-ct,t),\Psi(x-ct,t) \Bigr) = \int_{-\infty}^\infty |\Psi(x-...
0
votes
5answers
289 views
Confusion about ket states and bra with position
I am very confused about the bra-ket notation of states and the fact that
$$\psi(x) = āØx|\psiā©$$
and
$$āØx|x'ā© = \delta(x-x')$$
are true. What does this mean? What is the ket $|xā©$, is it just some ...
4
votes
1answer
522 views
Dirac Delta Function and Position [duplicate]
How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle xā|x\rangle = \delta(xā-x)$?
1
vote
0answers
88 views
Why does the integration over momentum has normalization constant of volume?
If I Fourier transform a wave function in position space, integration carries no normalization constant:
$$\displaystyle{\phi(k) \equiv \langle k|\psi\rangle = \sum\limits_x \langle k|x\rangle\langle ...
0
votes
1answer
106 views
Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate]
In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by
$$I = \sum_i |x_i\rangle\langle x_i|.$$
To me, ...
1
vote
0answers
85 views
Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]
I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions".
Given a finite dimensional vector space with a basis $\{x_i\}$, I understand ...
0
votes
2answers
186 views
Linear combination of 2 spherical harmonic functions
The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?
0
votes
1answer
257 views
Need for normalization in non-degenerate perturbation theory
I'm currently taking a class in QM and we came across the topic of non-degenerate perturbation theory. Let us for further discussions assume that $H_0$ is the unperturbed Hamiltonian with solutions of ...
1
vote
1answer
147 views
Ladder operators and energy levels [closed]
I am studying how to get the normalization factor algebraically in the exited states of the harmonic oscillator using the beautiful ladder operators technique.
I am stuck at the point where is stated ...
0
votes
0answers
65 views
Why are periodic functions through space disregarded in quantum mechanics?
My question is a general question arising from an observation from the wavefunction of a free particle.
One must disregard, for a free particle, that the spatial wavefunction is:
$$\psi(x) = \sum_n ...
-1
votes
1answer
139 views
Considering an arbitrary wavefunction for a free particle, are all normalizable functions valid?
One can show that a possible solution to a wavefunction with constantly zero potential is equal to, only considering the spacial piece:
$$\psi(x) = \int_{-\infty}^{\infty} A(k) \ e^{ikx} dk$$
This ...
