Questions tagged [normalization]

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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 $\omega$ ...
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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)}\...
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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 ...
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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 ...
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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 ...
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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 $\...
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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\...
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1answer
88 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 $N$...
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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 ...
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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 ...
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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 ...
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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&...
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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 ...
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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 $\phi$ ...
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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}) =...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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$...
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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 ...
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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 ...
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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?
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $U(1)_Y$ ...
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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[...
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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 ...
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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 ...
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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 ...
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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: ...
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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-...
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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 $$\psi(x) = ⟨x|\psi⟩$$ and $$⟨x|x'⟩ = \delta(x-x')$$ are true. What does this mean? What is the ket $|x⟩$, is it just some ...
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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 $\langle x’|x\rangle = \delta(x’-x)$?
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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 ...
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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, ...
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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 ...
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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 $1/sqrt(2)$?
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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 ...
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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 ...
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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 ...
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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 ...

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