Questions tagged [normalization]
The normalization tag has no summary.
370 questions
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Is it valid to keep $k_1$ and $k_2$ when considering a light propagating along $z$ axis?
If a mode function of the light is given by $\psi_{\mathbf k}(x^\mu)=ce^{ik_\mu x^\mu}$, where the degrees of freedom of polarization are suppressed, it can be normalized by requiring $\left <\psi_{...
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Why does this condition hold for normalizable wave functions? [duplicate]
I am reading Griffiths' Introduction to Quantum Mechanics, and on Page 14 the footnote states that in 1D normalizable wave functions $\Psi(x,t)$ goes to zero faster than $\frac{1}{\sqrt{|x|}}$, as |$x$...
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Units of wave functions in real and reciprocal space
I'm confused about the units of wave functions in reciprocal space and their Fourier transform in real space. On one hand, I believe the Fourier transform of a reciprocal space wave function in 2D is ...
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Question on the square-integrability of the given wavefunction at origin and infinity
I have this function as a wavefunction of a quantum system:
$$\psi(r)=N r^a \exp\left(br^2 + cr+\frac d{r^3}+\frac e{r^2}+\frac f{r}\right)$$
where $r$ is the radial parameter ranging on the interval $...
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Why are there constants in the Lagrangian? [duplicate]
I have a question regarding the constant terms in the lagrangian in field theory. Lets take the electromagnetic action in a vacuum for example:
$$
S = \int \left( -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} \...
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In perturbation theory, are there two or three summation terms in the second-order correction to the eigenfunctions?
Context
This question is a narrow one and it is specifically related to non-degenerte, time-independent perturbation theory.
In working through [1], Sakurai offers in Eq. (5.1.44) that the second-...
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Vacuum Polarisation Graphs Cancellation Theorem
I'm going through Mahan's "Many-Particle Physics", and I'm a bit confused about a theorem that he states about the vacuum polarisation terms cancelling out the terms with disconnected ...
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Can we no longer predict the behavior of a particle with a definite position?
This might be a really dumb question as I am just learning QM for the first time.
Shankar says that physically interesting wavefunctions can be normalized to a unit $L^2$-norm:
$$\int_{-\infty}^{\...
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Zero-point connected correlator $\langle 1 \rangle_C$ is not 1?
This is so confusing: books are saying that connected correlator is given by
$$\langle \phi(x_1) \phi(x_2) ... \phi(x_n) \rangle_C = \left.(-i)^{n-1}\frac{\delta }{\delta J(x_1)} \frac{\delta}{\delta ...
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Dirac-Delta from Normalization of Continuous Eigenfunctions
I'm following this paper, and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy
\begin{equation}
\langle f_s|f_{s'}\rangle = \int_{0}^{1} \...
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2
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Why do we write in general certain eigenfunctions with constants when the weights depend on the hamiltonian?
As a matter of habit, I've simply written out eigenfunctions of spin systems, say, with the usual normalization constants as weights, but now I'm being asked to write them in terms of Hamiltonian ...
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Quantum Mechanical Current Normalisation
Consider an electron leaving a metal. The quantum-mechanical current operator, is given (Landau and Lifshitz, 1974) by
$$
j_x\left[\psi_{\mathrm{f}}\right]=\frac{\hbar i}{2 m}\left(\psi_{\mathrm{f}} \...
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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 ...
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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 ...
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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 $\phi$ ...
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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 ...
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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 }+...
3
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1
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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 ...
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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:
$$
\hat p = \hbar^2 \int dk |k\rangle k \langle k|
...
2
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Normalization in the Abelian Chern-Simons action
In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
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Confusion Regarding the Propagator [duplicate]
To my understanding, the expression $$G^+=\theta(t_f-t_i)\langle x_f|\mathcal{\hat U}(t_f,t_i)|x_i\rangle$$ represents the probability amplitude that a particle starting at position $x_i$ at time $t_i$...
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Meaning of Scattering-Amplitudes "time-volume"
I've been reading this book by Schwichtenberg as I re-learn QFT. My question is general, but applies easiest to the scattering amplitude at first order in the $\phi^4$-theory for $k_1,k_2\rightarrow ...
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Problematic Factor of 2 in Klein-Gordon Propagator Derivation
I want to derive the Klein-Gordon Green function equation
$$(\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-...
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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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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 ...
