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Quantum Mechanical Current Normalisation

Consider an electron leaving a metal. The quantum-mechanical current operator, is given (Landau and Lifshitz, 1974) by $$ j_x\left[\psi_{\mathrm{f}}\right]=\frac{\hbar i}{2 m}\left(\psi_{\mathrm{f}} \...
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About momentum states covariant normalization

I'm following QFT of Schwrtz and I have a doubt about Eq. (2.72). In particular, from Eq. (2.69): $$[a_k,a_p^\dagger]=(2\pi)^3\delta^3(\vec{p}-\vec{k}),\tag{2.69}$$ and Eq. (2.70): $$a_p^\dagger|0\...
Albus Black's user avatar
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1 answer
87 views

What is the physical meaning of the normalization of the propagator in quantum mechanics?

Suppose we have a quantum field theory (QFT) for a scalar field $\phi$ with vacuum state $|\Omega\rangle$. Then, in units where $\hbar = 1$, we postulate that the vacuum expectation value (VEV) of any ...
zeroknowledgeprover's user avatar
1 vote
1 answer
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Confusion on Shankar's Motivation for the Dirac delta Function

I was reading Shankar's Principles of Quantum Mechanics and got confused on page 60, where he motivates the delta function from the normalization problem of the inner product for function spaces. We ...
Han's user avatar
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Discrete to continuous quantum operator

Let's say that we have a discrete lattice with $N$ sites. Let's label the site by the index $i$. Let's say that we have the operators $a_i$ and $a_i^\dagger$ which correspond to the creation and ...
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Homogeneity of Schroedinger equation implies norm conservation

I am trying to understand how homogeneity of Schroedinger equation implies norm conservation. I know that we are considering the non-relativistic case, where particle number is conserved, so we do not ...
imbAF's user avatar
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1 answer
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Square Integrability of spherical symmetric wave

In my class we were discussing some wave equation for a spherical symmetric wave $u(t,r)$ and my professor investigated the behaviour of the solutions asymptotics $r \rightarrow \infty$. The solution ...
Octavius's user avatar
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Does the inner product of wavefunctions really have units? [closed]

Let $\psi(x)$ and $\phi(x)$ be wavefunctions. I usually see the inner product defined as $$\int dx\, \overline{\psi(x)} \, \phi(x)$$ and interpreted, I think, as "the amplitude that state $\phi$ ...
Upasker's user avatar
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Angular momentum completeness relation

Can anyone tell me if the angular momentum completeness relation is given by $$ \sum_{l=0}^{\infty} \sum_{m=-l}^{l} (2l + 1) |l,m\rangle\langle l,m| = I $$ or $$ \sum_{l=0}^{\infty} \sum_{m=-l}^{l} |l,...
Dr. user44690's user avatar
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Continuum bases: why do we use dirac delta function? [duplicate]

In discrete bais, we can express a vector as $$ |\psi\rangle=\sum_{i} c_i|e_i\rangle $$ with orthonormality $$ \langle e_i|e_j\rangle=\delta_{ij}.$$ $\delta_{ij}$ is usual kronecker delta. If we ...
B. Silasan's user avatar
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What is the difference between $(\mathcal{H}\setminus \{ 0\})/\mathbb{C}^*$ and $\mathcal{H}_1/U(1)$?

Let $\mathcal{H}$ be a Hilbert space. We define the projective Hilbert space $\mathbb{P}\mathcal{H}$ as $\mathcal{H}\setminus \{ 0\}/\mathbb{C}^*$. Then $[\Psi]=\{ z\Psi :z\in \mathbb{C}^*\}$. On the ...
Mahtab's user avatar
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How do you determine that the series solution to the hermite differential equation is not square integrable?

When solving the Schrodinger equation of the harmonic oscillator in one dimension you encounter the hermite differential equation: \begin{equation} \left[\frac{d^{2}H}{d\xi^{2}}-2\xi\frac{d H}{d \xi }+...
Gueladio KANE's user avatar
1 vote
1 answer
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Calculating the average kinetic energy (expectation value) of gas particles from the Maxwell Boltzmann distribution

From what I already know, to calculate the expectation value/average from a probability distribution, you use the formula: $$ \langle x \rangle \ = \int_{-\infty}^{\infty} x f(x) \,\mathrm{d}x \tag{1}$...
user374355's user avatar
3 votes
1 answer
131 views

How can the linear combination of infinite normalized Klein-Gordon fields be a normalizable field?

In the context of a Klein-Gordon field with charge $e$, mass $m$, immersed in an external classical electric field $A_\mu = (A_0(z), 0)$, I am asked to calculate the charge density of the field ...
dolefeast's user avatar
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1 answer
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Differential Cross Section and Factor of $\pi$

I hope this is not a double-post, but the other threads couldn't help me: In my calculations of the differential cross section $\frac{d\sigma}{d\Omega}$, I am always a factor $\pi$ lower than the ...
MCSquared's user avatar
1 vote
2 answers
98 views

Changing the basis from $p$-basis to $k$-basis in standard quantum mechanics

if $$\hat p = \int dp |p\rangle p \langle p|$$ and I want to chage the basis to $|k\rangle$ it is correct to say that $\hat p$ is therefore equal to: $$ \hat p = \hbar^2 \int dk |k\rangle k \langle k| ...
TheWhitelily2010's user avatar
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1 answer
122 views

Proof of normalising the Dirac spinors

I was reading through my particle physics textbook and saw a property of the Dirac spinors that I did not understand. The spinor is defined by $u^s(p)=\sqrt{\frac{E+m}{2m}} \begin{bmatrix}\phi^s \\ \...
Chris G's user avatar
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Normalization in the Abelian Chern-Simons action

In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
Tuhin Subhra Mukherjee's user avatar
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2 answers
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Normalization to unity, Projection Operators in QM

I have a question about something that is stated in Sakurai's MQM. It's written that if one runs a sequence of selective measurements (namely, a sequence of independent stern-gerlach apparatuses) ...
Claudio's user avatar
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0 answers
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Confusion Regarding the Propagator [duplicate]

To my understanding, the expression $$G^+=\theta(t_f-t_i)\langle x_f|\mathcal{\hat U}(t_f,t_i)|x_i\rangle$$ represents the probability amplitude that a particle starting at position $x_i$ at time $t_i$...
Diego Ramil's user avatar
2 votes
1 answer
192 views

Normalization of the harmonic oscillator propagator

The propagator of a quantum system is defined by $$\mathcal{K}(t,x;\,t_{0},x_{0})\,\equiv\,\left\langle x\right|\hat{U}(t,\,t_{0})\left|x_{0}\right\rangle.$$ In this notation, the unitarity demands ...
Yair's user avatar
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1 vote
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Meaning of Scattering-Amplitudes "time-volume"

I've been reading this book by Schwichtenberg as I re-learn QFT. My question is general, but applies easiest to the scattering amplitude at first order in the $\phi^4$-theory for $k_1,k_2\rightarrow ...
derEddi's user avatar
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2 votes
1 answer
139 views

Free particle probability to go from $a$ to $b$ [duplicate]

Feynman and Hibbs write that the probability for a particle to go from $a$ to $b$ is \begin{equation*} P(b,a)=|K(b,a)|^2 \end{equation*} The kernel for a free particle is given as \begin{equation*} K(...
ipie's user avatar
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1 vote
0 answers
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Problematic Factor of 2 in Klein-Gordon Propagator Derivation

I want to derive the Klein-Gordon Green function equation $$(\Box_b + m^2) D_F(x_b - x_a) = - i \delta^4(x_b - x_a)$$ by using the same steps taken when fixing the 'exact' Green function of the non-...
bolbteppa's user avatar
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1 vote
1 answer
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Sublattice Magnetization of Heisenberg Model on triangular lattice

I'm trying to rederive the expressions for the square of the sublattice magnetization of the $120^\circ$ Neel antiferromagnetic (NAFM) and the stripe phase on the triangular lattice as shown in ...
WikawTirso's user avatar
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1 answer
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Is unitarity equivalent to linearity plus conservation of the norm?

Unitarity is the condition that the inner product in the Hilbert space is preserved. But if you suppose that the norm of any state is already preserved, then does unitarity follow from linearity? ...
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1 answer
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Expectation Value Definition [closed]

I think I might just not be thinking things completely through here but I've seen the expectation value of a Hermitian operator written in many sources as both $$\langle A \rangle=\frac{\langle\psi|A| ...
d0uble_a_b4ttery's user avatar
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1 answer
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Normalisation and Dirac's Formulation of the Path Integral

Zee's Quantum Field Theory in a Nutshell (Dirac's Formulation in Chapter 1.2) contains the following passage (in attached image). Can someone please explain where the normalization is used in getting ...
Jake McNaughton's user avatar
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1 answer
45 views

Two normalization constants for Dirac plain wave function

I stumbled across two different expressions for a Dirac plane wave function, namely $$\psi=\sqrt{\frac{m}{EV}}ue^{-ip\cdot x}$$ and $$\psi=\frac{1}{\sqrt{2EV}}ue^{-ip\cdot x}$$ where $u$ is the Dirac ...
hendlim's user avatar
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3 votes
1 answer
178 views

How to normalize the states in the continuous limit?

In quantum field theories we can perform the continuous limit, where we take the limit $V\rightarrow\infty$ for the volume system. In quantum optics, we can start by absorbing a factor $\left(\frac{L}{...
HerpDerpington's user avatar
5 votes
1 answer
172 views

Non-normalizable eigenfunctions

I was reading Shankar's Principles of Quantum Mechanics when on page 65, he starts talking about infinite spaces and operators in them. He introduces an operator $K$, which in the $x$ basis takes the ...
JS30's user avatar
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1 answer
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I don't understand the normalization of a quantum state in quantum machine learning paper

In the paper arXiv:1307.0411 by Seth Lloyd, Masoud Mohseni, Patrick Rebentrost on page 3, line 8, it says: Constructing the $\log_2N$ qubit quantum state $| v \rangle = |\vec{v}|^{-1/2}\vec{v}$ then ...
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1 answer
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How to normalize the symmetrizer of a qudit state?

Notations : $\mathcal{H}= \mathbb{C}^d \otimes \cdots \otimes \mathbb{C}^d$ where $\mathbb{C}^d$ appears $N$ times. $S(\mathcal{H})$ is the symmetric subspace of $\mathcal{H}$. $(|\epsilon_1\rangle,...
Baloo's user avatar
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2 votes
0 answers
51 views

Normalizability of thermal density matrix in QFT

Suppose that I have a system described by a quantum field theory with Hamiltonian $H$ and $U(1)$ charge $Q$. The thermal density matrix is then $$ \rho = \frac{1}{Z} e^{-(H-\mu Q)/T},\quad Z=Tr(e^{-(H-...
user105620's user avatar
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2 votes
2 answers
172 views

Why rescale the kinetic term in Wilsonian renormalization?

I have been doing some reading on Wilsonian renormalization and also Effective Field Theories. It's my understanding, and I could be wrong, that part of the process is to continually rescale the ...
Cory's user avatar
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-1 votes
3 answers
152 views

Why exactly $\langle x' | x \rangle = \delta(x'-x)$? [duplicate]

If the position state $|x\rangle$ is complete and orthonormal I understand that $$\langle x | y \rangle = \delta(x-y).\tag1$$ However, why exactly $$\langle x' | x \rangle = \delta(x'-x)\tag2$$ How do ...
InvisibleParticle's user avatar
1 vote
1 answer
139 views

What happens to the normalization condition if the wave function is non stationary?

The quantum mechanics courses I have taken as of now pretty much only deal with stationary states of the form: $$\psi(x,t) = \psi(x)\exp{\left(-\frac{iE}{\hbar}t\right)} $$ Now, what about situations ...
agaminon's user avatar
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1 vote
0 answers
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Expression of Dirac delta function by integral of exponential function [duplicate]

In many QFT books including Peskin & Schroeder's and M.Schwartz's mention about Fourier transform and representation of Dirac delta function as $\begin{align} &f(x) = \int\frac{d^3p}{(2\pi)^3}...
hwan's user avatar
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4 votes
2 answers
341 views

When do we exclude non-normalizable solutions and when not?

I'm a bit confused on when we should keep any non-normalizable solutions and when not. What do I mean? Let's say that we have the free particle system. The energy eigenstates are not normalizable - ...
MTYS's user avatar
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1 answer
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Normalized units versus dimensionless units

In a molecular dynamics code, suppose, the distances are expressed in units of a characteristic length of the simulated system, $R_0$. In some papers it is written as, " distances are normalized ...
bubucodex's user avatar
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1 answer
130 views

How to normalize a wave function for different potential? [closed]

I have 3 different wave functions for 3 different potentials. Basically, this is the intereaction for a diatomic molecule. $$\psi_1(x) = 0, x<0$$ $$\psi_2(x) = C^{ibx} - C^{-ibx}, 0 \le x \le a$$ $$...
happypaticle's user avatar
0 votes
1 answer
47 views

How do you use Boltzmanns distribution law?

Considering a particle in an isothermal atmosphere: $ f \left( h \right) \delta \left( h \right) = A e^{\frac{-m g h}{k T}} \delta \left( h \right)$ where $A$ is the normalisation constant of the pdf. ...
thicccjk's user avatar
4 votes
1 answer
273 views

Wavefunction renormalization $Z_n$ in QM perturbation theory given in J.J. Sakurai's book

I'd been studying introductory quantum mechanics from J.J. Sakurai's Modern Quantum Mechanics, 3rd Edition. I was reading chapter 5, especially about time-independent perturbation theory, non-...
GeorgePhysics's user avatar
2 votes
1 answer
368 views

Peskin and Schroeder's QFT eq. (9.14): Gaussian momentum field integration of phase space path integral

On Peskin and Schroeder's QFT book page 282, the book considered functional quantization of scalar field. First, begin with $$\left\langle\phi_b(\mathbf{x})\left|e^{-i H T}\right| \phi_a(\mathbf{x})\...
Daren's user avatar
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3 votes
1 answer
231 views

Measure of Functional Integral in Path Integral Formulation

I have a question regarding the prefactor $\sqrt{\left(\frac{m}{2\pi i \hbar \Delta t}\right)}$ in $$\left<x'|e^{-iHt}|x\right> = \int D[x] \exp(\frac{i}{\hbar}\int dt' L(x, \dot{x})),$$ where $$...
Neophyte's user avatar
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2 votes
1 answer
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Why two energies look same in the relativistic normalization?

I'm reading Peskin's QFT textbook. In this book, to make normalization of momentum eigenstate Lorentz invariant, we define momentum eigenket as $$\left| \mathbf{p} \right> = \sqrt{2E_{\mathbf{p}}} ...
김승현's user avatar
-1 votes
1 answer
498 views

What is the difference between Dirac delta function orthogonality and Kronecker delta orthogonality?

In the derivation of Bloch Wave, I encountered a problem. First of all this is the definition of Bloch Wave: $$ \psi_{n\mathbf{k}} (\mathbf{r} ) = e^{i\mathbf{k} \cdot \mathbf{r} } u_{n\mathbf{k}} (\...
Zhao Dazhuang's user avatar
2 votes
0 answers
104 views

(Srednicki) How to obtain the normalization condition for Dirac field?

I'm reading through srednicki's qft and I met a problem. In its section 41, after he make an assumption that the creation operators of free field theory would work comparably in the interacting theory ...
Dzhou's user avatar
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0 answers
30 views

Chemical potential for BE condensate: why the normalization would (uncorrectly) give $\mu> 0$?

Consider a gas of bosons. Let $g(\epsilon)$ be the distribution of energy levels (degeneracy). Consider the following integral $$I(\beta,\mu)=\int{d\epsilon g(\epsilon}) \frac{1}{e^{\beta(\epsilon-\mu)...
Simon's user avatar
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-1 votes
1 answer
102 views

Normalize the wave function with spherical harmonics [closed]

I have this wave function: $$\Psi=C e^{-\rho / 2} \rho^{l} L_{1}^{3} Y_{l, m} $$ To normalize the function I have tried to express the polynomial in the function as follows. If: $$L_{1}^{3}(x)=(-1)^{1}...
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