Questions tagged [normalization]

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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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Mathematics required for understanding quantum operators and the preservation of wave-function's norm [closed]

I am currently in High school and doing the IB curriculum. I just finished with my first year and moving on to my final year.As part of my highschool diploma, I have to complete an essay on any ...
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2answers
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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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Orthonormality of Hermite function [migrated]

I was wondering if someone could tell me when the following relation holds? where $H_{n}(x)$ are Hermite polynomials and $\delta(x-x')$ is dirac delta function. $$ \sum_{n=0}^\infty \frac{1}{\sqrt{\pi}...
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1answer
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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
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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
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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
97 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
105 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
46 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
68 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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'Show that $E$ must exceed $V(x)$ for every normalisable solution of the TISE'. Surely you can show that this isn't true using exponentials?

In Griffith's this is a question, there's the clear answer which jumps out immediately: $$\frac{d^2\psi}{dx^2} = \frac{2m}{\hbar^2}(V(x)-E)\psi.$$ That is, if E is less than V the second deriviative ...
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1answer
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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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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
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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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217 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
68 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
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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
111 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
66 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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39 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
64 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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1answer
50 views

Can this wave function be normalized? [closed]

This question I am stuck on goes like this: The ground state wave function for the electron in a hydrogen atom is $c\ e^{-r/a}$ where $r$ is the radial coordinate of the electron, $c$ is a constant ...
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1answer
80 views

Pion decay constant: How to know which convention to follow?

As summarized by Wikipedia, different sources use different choices for the (pion) decay constant. This means that the numerical value can vary between $$ \sqrt 2\ f_\pi \quad\leftrightarrow\quad f_\...
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1answer
121 views

How is it possible to take the inner product of states which belong to two different Hilbert spaces?

Question 1 In case of spontaneous breakdown of a continuous symmetry e.g. the ${\rm U(1)}$ symmetry, two different vacua can be labelled as $|\theta\rangle$ and $|\theta^\prime\rangle$, and they ...
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Why does $\langle x' | x \rangle$ give the Dirac delta distribution? [duplicate]

I'm having difficulty understanding why the following is true: $$ \int_\mathbb{R} \langle x' | x \rangle dx = \int_ \mathbb{R} \delta(x-x')dx$$ where $\delta(x)$ is the delta distribution. Are we ...
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1answer
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Pöschl–Teller free wave solution normalization

I'm considering one-dimentional QM ($\hbar=1$, $m=1$) with the following potential $$ V(x) = - \frac{1}{\cosh^2 x}\;. $$ I know that free-wave solution are $$ ψ_k(x) = e^{\pm i k x}(\tanh x \mp ik)\;. ...
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1answer
33 views

Existence of normalizable solution in specific system

Suppose we have some function $\Psi(x,y)$ satisfying \begin{align} &\partial_{x} \partial_{y} \Psi(x,y)=0 \\ &(\partial_{x}^2 - \partial_{y}^2) \Psi(x,y)=0. \end{align} Then how can we ...
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1answer
277 views

What is the general form of the wave function? [closed]

Questions: What is the general form of the wave function, i.e. $\psi$ in Schrödinger's equation? What is the general form of the integrated wave function, i.e. $\operatorname{Eq.}{\left(1\right)}$?$$ ...
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Are eigenfunctions always normed and orthogonal?

I came across this simple proof: We show that Hermitian operators have real eigenvalues. The definition of a Hermitian operator is \begin{equation} \langle \phi_i | \hat A | \phi \rangle = \...
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273 views

Yang-Mills action - a potential mistake in Wikipedia

Currently at the Wikipedia page on Yang-Mills theory, you see that [a screenshot], Isn't that an obvious mistake? Based on this normal convention: $$ F^2 \equiv F \wedge \star F $$ Shouldn't ...
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156 views

Wave-Function Normalization in Momentum Space Not Possible

Hello, I just have a question about this passage; specifically, I do not understand why the result of the inner product (the integral of u_k* and u_k') being the delta function defies conventional ...
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3answers
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Why do some solutions of the Schrodinger's equation emerge un-normalized? It goes against logic

Why do some solutions of the Schrodinger equation emerge un-normalized? Logically, any solution psi must have its integral $\int_{-\infty}^\infty |\varphi|^2\, dx = 1$ because the probability has to ...
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1answer
66 views

Normalization of Probability distribution [closed]

I need to Know. Is it a condition that Probability density is bounded between 0 and 1?
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1answer
72 views

Transition from an initial/final position state to the ground state in the path integral

I am reading chapter 6 of M.Srednicki's book. On page 47 he argues why it is possible to choose for the initial/final state the ground state instead of a position state. Actually I don't understand ...
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1answer
74 views

Vector space in quantum mechanics

Do the set of all square-integrable normalized functions necessarily form a vector space in quantum mechanics? The reason for this question being my problem in not understanding why the zero is not ...
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1answer
72 views

Confusion in normalization of position space and momentum space

It seems that in LSZ formalism approach, or just Feynman diagram approach, we can compute scattering amplitude of $\langle x_{out} | y_{in}\rangle$ (position space) and $\langle p_{out} | p_{in}\...
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1answer
116 views

Feynman path integral normalisation from completeness condition

The path integral for some potential can be evaluated explicitly by discretion the space and performing $N$ Gaussian integrals then taking the limit as $N \to \infty$ for the case of a free particle. ...
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Hydrogen Atom, polar equation eigenfunctions

In my textbook, Quantum Mechanics by David McIntyre on page 235, the solutions to the polar equation resulting from the separation of variables to the hydrogen atom are the eigenstates: The book ...
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1answer
59 views

What is the scope of the term 'normalisation'?

When we 'normalise' the wavefunction we put in an appropriate coefficient so that the wavefunction can act as a probability distribution. However, when I considered the eignefunctions of the momentum ...
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1answer
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How to proceed with dimensionless recasting of telegrapher's equation?

I am referring to this paper, on page 3 it is given that: The telegrapher's equation is given as $$\partial_t P_+=D\partial_x^2P_+-v\partial_xP_+-\gamma P_++\gamma P_-,$$ $$\partial_t P_-=D\...
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1answer
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Why do we consider vectors of length $N = 2^n$ for the Quantum Fourier Transform?

Discrete Fourier Transform Classical Discrete Fourier transform acts on a vector $(x_0, x_1, ..., x_{N-1}) \in C^N$ and maps it to vector $(y_0, y_1, ..., y_{N-1}) \in C^N$ according to the formula $$...
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Two-Point Correlators in Conformal Field Theories

In a CFT we normalize the two-point correlators for two fundamental scalars as $\langle\phi(x)\phi'(y)\rangle = \frac{\alpha\delta_{\phi\phi'}}{\left|x - y\right|^{2\Delta}}$ , where $\alpha$ is a ...
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551 views

How to prove an expression is not a solution in 1D Schrödinger equation? [closed]

I was reading Griffith's book Introduction to Quantum mechanics and found that for the case of a free particle, we can diregard solutions of the form $e^{kx}$, where $k$ is real (positive or negative)....
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1answer
68 views

Continuous limit of discrete position basis

Say we have a $1D$ lattice with spacing $a$ between two sites. How does one formally map the discrete position basis of the lattice to a continuous one in the limit $a\to 0$. For instance how does ...
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295 views

Obtaining momentum space wave function

Suppose a wave function $\psi (x)=C\exp(ip_0x/\hbar)\exp(-|x|/(2\Delta x)) $ is given and we are asked to find the momentum space wavefunction. Then we take the Fourier transform as follow: \begin{...
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How to normalize a non-integrable function? [closed]

Given is $$ \Psi(x,t) = N \exp \left(-ax^2 + ibt \right) $$ On the particle acts a force which is: $$ F = -kx$$ that means that the potential is: $$V = \frac{1}{2}kx^2$$ the Time-Dependent ...