Questions tagged [normalization]

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Normalization of solution of Dirac equation

I know that the solution to the dirac equation are of the form: $\psi(x)=u(\vec{p})e^{ip\cdot x}$ and the spinor can be normalized as $u^\dagger u =E$. I was reading "Lectures on Quantum Field ...
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How would I normalize this ket vector? [closed]

So I am given the vector: $$|Ψa⟩ = |x⟩ + |y⟩ − |z⟩$$ And I need to normalize it. I know that I have to take the dot product of the vector with itself (and it needs to equal 1) but how would I do this ...
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5 votes
2 answers
263 views

A common standard model Lagrangian mistake?

A common standard model lagrangian is written in a cup like this. It appears in many places also on a T shirt. But isnt that there is an obvious mistake? That the Dirac lagrangian is already itself ...
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How to normalize a two-particle state?

Say I have a state of two non-interacting fermions in some system, $$\Psi_{12}(x_1,x_2)=\frac{1}{\sqrt{2}}(\Psi_1(x_1)\Psi_2(x_2)+\Psi_1(x_2)\Psi_2(x_1))\otimes\frac{1}{\sqrt{2}}(\uparrow\downarrow-\...
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Probability for scattering event

I am reading Schwartz QFT. On page 61 in eq (5.20) he gives an expression that describes the probability for a $2\to n$ scattering event to happen: $$dP=\frac{T}{V}\frac{1}{(2E_1)(2 E_2)}\left|\...
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Normalising a free particle wave function, at $t=0$

I am trying to normalise the wave function $\psi$ for a free particle, with initial boundary conditions. $$\Psi(x,0)=Ae^{-2|x|}.$$ When trying to normalise it, I keep getting $\infty$ which clearly ...
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3 answers
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Questions about normalizing wavefunctions [closed]

learning QM and just have a few questions regarding normalizing wavefunctions. Thus far, every initial wavefunction that we've normalized has had an undefined constant explicitly put out front (i.e., ...
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2 votes
1 answer
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Normalizable, but singular distribution

I have obtained a probability distribution for the observable $l$ which takes the form: $$ \frac{dP}{dl}=\frac{(1-\sqrt{1-3l^{2}})^{2}}{l^{3}\sqrt{1-3l^{2}}}\exp\left[-\frac{4\pi}{9l^{2}}(1-3l^{2})^{3/...
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Kohn-Sham equations, Sakurai 3rd edition, possible typo?

In Sakurai's quantum mechanics book 3rd edition page 448, equation 7.88, the book writes "Kohn and Sham found a way to derive a self-consistent approximation scheme, based on single particle ...
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1 answer
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Showing that a wavefunction in column form is normalised [closed]

I am given the following wavefunction in column form: $\psi = \begin{bmatrix} \frac{1}{4} \\ \sqrt{\frac{15}{16}}i \end{bmatrix} $ And asked to show that it is normalised. As I understand it, the ...
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Why do we need to normalise states in quantum field theory?

In QM its obvious that we need to normalise quantum states since their inner product squared represents a probability. This normalization leads to physical states in QM being represented by 'rays' of ...
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When do we normalize a wave equation? In what kind of problems/exercises? Why?

I'studying quantum mechanics, and i haven't understand very well, when should we normalize the wave equation? And why must we normalize it?
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2 answers
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Factor $1/\sqrt{2\pi}$ in the normalization of wave function packet

My book has started using the wave packet definition as follows (time independent form): $$\Psi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} A(k) \ e^{ikx}dx$$ I do not understand where the $1/\...
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2 votes
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Normalization in vector field in QFT after non-relativistic expansion

I encountered this equation when I was reading the article "Black Hole Superradiance Signatures of Ultralight Vectors" $$A_\mu=\frac{1}{\sqrt{2m}}\Big(\psi_\mu (\vec{r})\exp(-i\omega t)+\...
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The Density Matrix of a Pure State

It is my understanding that any pure quantum state $|\psi\rangle$ can be represented by the density matrix $|\psi\rangle\langle\psi|$. It is also my understanding that $|\psi\rangle\langle\psi|$ ...
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Probability of non-normalizable states

In the book Quantum Field Theory by Jakob Schwichtenberg, he is discussing about non-normalizable states in chapter 8. If you compute the normalization in $(8.67)$ you just get infinity. He said one ...
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Proving that the normalization is independent of time

The function that I want to normalize represents an Airy wave-packet: $$\psi(x,t)=\mathrm{Ai}[q(x-ut+ivt-\tfrac12at^2)]e^{i\frac{mat}{\hbar}(x-ut-\frac13at^2)}e^{\frac{mv}{\hbar}(x-ut+\frac i2vt-at^2)}...
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1 answer
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Normalization problem of periodic wave function

So, I want to normalize the eigen wavefunctions of the momentum operator ($-i\hbar \frac{\partial}{\partial x}\psi(x)=p\cdot\psi(x)$ where $p$ is a real number). The solution is $\psi(x)_p=C\cdot e^{\...
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What is $\langle 0|p\rangle$?

$\hat{p}$ is the generator of the translation group, so $$|r\rangle=e^{-ir\hat{p}/\hbar}|0\rangle\to\langle p|r\rangle=e^{-irp/\hbar}\langle p|0\rangle.$$ Assuming normalized position states \begin{...
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Factor $\frac{1}{2}$ in scalar kinetic Lagrangian in QFT [duplicate]

Why is it that sometimes I see kinetic term of scalar Lagrangians written like this $$\mathcal{L}=\partial_\mu\phi^\dagger\partial^\mu\phi+\dots$$ like for example in scalar electrodynamics, while ...
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2 votes
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Normalization of One-Particle States for Klein-Gordon Field Quantization

Peskin & Schroeder in their QFT textbook discusses how we may normalize one-particle states $|\textbf{p}\rangle$ for Klein-Gordon field quantization in pages 22-23. The excerpts are given below. ...
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Normalisation in Dirac Notation

Say I have a wave function as follows (example): $$|\psi\rangle=|\phi_1\rangle-\sqrt{3}|\phi_2\rangle+ 2i|\phi_3\rangle$$ I know normalisation means: $$\langle \Psi_N|\Psi_N \rangle =1\tag{1}$$ I know ...
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'Normalization' of free particle wave function [duplicate]

I'm trying to obtain the expression for the free particle that is known $$\psi_p(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{i\frac{xp}{\hbar}}$$ Easily you can arrive to the exponential, $$p\langle x|p\rangle=-i\...
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How to normalize a linear combination of spherical harmonics?

I know the formula for normalizing individual spherical harmonics, but do not know how to normalize linear combinations of them Say I have a system $ \alpha (\theta, \phi) = aY_{l_1}^{m_1} + bY_{l_2}^...
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Can $n$ be negative for the infinite square well wave function $\psi_n(x)$?

Consider the classic particle in a box example (infinite square well) in quantum mechanics: \begin{equation} \psi_n(x)=A\sin(k_n x), \end{equation} where \begin{equation} k_n=\frac{n \pi}{L}. \end{...
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Restricting the range of $x$ to normalize the wavefunction

I have a system in nuclear physics which can be approximated by two coupled harmonic oscillators (the Hamiltonian is a bit peculiar though.) I will denote their spatial coordinates by $x$ and $y$. I ...
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LSZ formula in Srednicki, normalization issue

In the Ch.5 of his book, Srednicki says LSZ formula is valid provided the following conditions hold: $$ \langle 0|\phi(x)|0\rangle = 0, \langle p|\phi(x)|0\rangle = 1 $$ To achieve these conditions, ...
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4 answers
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Using separation of variables to solve Schrödinger equation for a free particle

I was reading Introduction to Quantum Mechanics by David Griffiths and I am in Chapter 2, page 45. I know that since the solutions from Schrödinger equation cannot be normalized for a free particle. ...
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Weinberg's normalization convention for momentum eigenstates

In this answer https://physics.stackexchange.com/a/376193/274751 two different conventions for the normalization of momentum eigenstates are mentioned. This convention amounts to the choice of $N(p)$ ...
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The normalization of the momentum eigenfunction [duplicate]

If the momentum eigenfunction is this but it is not normalized, and if we apply the normalization condition which is this will you get infinity instead of 1?
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Floating Normalization of Experimental Data Sets while Fitting Multiple Models

I have $N$ data sets of unequal cardinality, and I am told we do not treat each data set with a normalization of $1$. Instead we let the the normalization float and fit it as though it were any other ...
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6 votes
2 answers
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Normalization of vacuum state in field theory

I am doing a calculation of an amplitude in QFT, not an expert in the subject so this may be a trivial question but cannot find the answer. What is the normalization of the vacuum state of the ...
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Eigenfunction of wave vector [closed]

I am reading some book, where it is said that the eigenfunctions are given by $$\langle \mathbf{r}|\mathbf{k}\rangle = \frac{1}{\sqrt{\Omega}} \mathrm{e}^{i \mathbf{k} \cdot {\mathbf{r}}}$$ First of ...
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2 answers
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Normalisation of the Gaussian wave packet

I have been trying to solve the question asking for the normalisation of the Gaussian wave packet's probability density given as $$\rho(x)=Ae^{-\lambda(x-a)^2}$$ The $\rho(x)$ is just the probability ...
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1 vote
1 answer
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Normalization of $U(1)$ gauge fields

In G. W. Moore, “Introduction to Chern-Simons theories.” 2019 TASI School. [Online]. Available: https://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf the $U(1)$ gauge field has a ...
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2 answers
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Defined Momentum vs. Defined $k$

In quantum mechanics usually we write the momentum operator $\hat{p}$ as: $$\hat{p} = \hbar \hat{k}. \tag{1}$$ with of course: $$\hat{p}|p\rangle = p |p\rangle \tag{2}$$ $$\hat{k}|k\rangle=k|k\rangle \...
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2 votes
1 answer
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Normalization of momentum eigenstates in QFT

Inspired by a previous question, I'd like to ask about the normalization of one-particle states in QFT. The most common normalization seems to be the covariant one: $$ \langle \vec p'|\vec p\rangle = (...
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How the normalization condition implies the following relation?

Using equation 2.35 from Peskin and Schroeder: $$ |\vec{p}\rangle=\sqrt{2 E_{\vec{p}}} a^{\dagger}_\vec{p} |0\rangle $$ should lead to $$ U(\Lambda)|\vec{p}\rangle = |\Lambda \vec{p}\rangle, $$ where ...
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2 votes
1 answer
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Effects of a Lorentz boost on the normalization of a probability density (Dirac equation)

My question regards the probability densities of the Dirac equation. As is well known, the Dirac equation implies a continuity equation $$ \partial_\mu j^\mu = 0 $$ for $j^\mu = c\overline\psi \gamma^\...
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3 answers
738 views

Proof of normalization constant of wave function to be independent of time

I am trying to prove that the normalization constant is independent of time. If we have fixed it for a particular time then it will remain constant for all time. Suppose $\psi(x,t)$ is a wavefunction. ...
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2 answers
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How do I normalise the wavefunction of a hydrogen 1s orbital to obtain the normalisation constant?

The wavefunction I've been given for a 1s hydrogen orbital is: $$ \Psi = A e^{-r} $$ And I need to normalize this to find the value of A. I understand to normalise this I would inset this wave ...
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What is meant by a probablity given by $e^{-\text{P.E.}/kT}$ with $\text{P.E.}<0$?

This is from https://www.feynmanlectures.caltech.edu/I_40.html Let us take the case of just two molecules: the $e^{-\text{P.E.}/kT}$ would be the probability of finding them at various mutual ...
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1 vote
1 answer
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Understanding the Path Integral Formulation

Currently, I am reading chapter 3 of Condensed Matter Field Theory, which is on the Path Integral formulation of quantum mechanics. The book denotes $\Delta t = t/N$, where $t$ is the total time ...
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3 votes
2 answers
405 views

Coefficients of the wave function - a free particle in a box [closed]

If we solve the time independent Schrödinger equation for a particle in a box of length $L$, we get: $$\psi_n\left(x\right)=A\sin\left(\frac{\pi n}{L}x\right)$$ I then see that we normalize $A$ such ...
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2 votes
1 answer
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$\int_{-\infty}^{\infty} |\psi(x)|^2 ~ dx = 1$ when $\psi(x) = C\exp\left(\frac{x^2}{2a^2} + \frac{ix^3}{3a^3}\right)$ [closed]

The information given is: Consider a state $|\psi\rangle $ describing a quantum particle on a line, whose position representation $\langle x|\psi\rangle = \psi(x)$ is given by: \begin{gather*} \...
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1 answer
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Normalization of Hamiltonian Eigenfunctions for Free Particle [closed]

I am trying to prove that given $$\phi_E(x) = \left(\frac{m}{2E}\right)^{1/4} \frac{1}{\sqrt{2 \pi \hbar}} e^{i \sqrt{2mE}x/\hbar}$$ Then, $$\int dx \phi^*_E \left(x\right) \phi_{E^\prime}(x) = \delta(...
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Glauber Surdarshan $P$ Representation Normalizaion

I am studying Scully and Zubiary's quantum optics book currently, and I ran across their definition of the P representation as: $$P(\alpha,\alpha^*) = Tr[\rho \delta(\alpha^* - a^\dagger) \delta(\...
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1 vote
1 answer
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Value of matrix elements between bound and unbound states

How do you assign a numerical value to the matrix element between states of the discrete part $|n\rangle $and the continuous parts $|\alpha\rangle $of a spectrum of an operator ? The states of the ...
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1 answer
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Normalization of a wavefuntion [closed]

I am working with the following wavefuntion which describes two entangled photons. I need to normalize it over the frequency domain, $\omega_\alpha$ and $\omega_\beta$ are the frequency of the ...
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1 answer
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Normalization relation of the generators of the Lie-algebra of the Yang-Mills gauge group

At the introduction to Yang-Mills-theory and its gauge group typically a $SU(N)$-group, the generators $t_A$ of the corresponding Lie-algebra are supposed to fulfill the following normalisation: $$Tr(...
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