Questions tagged [normalization]
The normalization tag has no usage guidance.
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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\...
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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 ...
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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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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 ...
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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 ...
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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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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
$$
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} |l,...
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Continuum bases: why do we use dirac delta function? [duplicate]
In discrete bais, we can express a vector as
$$ |\psi\rangle=\sum_{i} c_i|e_i\rangle $$
with orthonormality
$$ \langle e_i|e_j\rangle=\delta_{ij}.$$
$\delta_{ij}$ is usual kronecker delta. If we ...
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What is the difference between $(\mathcal{H}\setminus \{ 0\})/\mathbb{C}^*$ and $\mathcal{H}_1/U(1)$?
Let $\mathcal{H}$ be a Hilbert space. We define the projective Hilbert space $\mathbb{P}\mathcal{H}$ as $\mathcal{H}\setminus \{ 0\}/\mathbb{C}^*$. Then $[\Psi]=\{ z\Psi :z\in \mathbb{C}^*\}$.
On the ...
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How do you determine that the series solution to the hermite differential equation is not square integrable?
When solving the Schrodinger equation of the harmonic oscillator in one dimension you encounter the hermite differential equation:
\begin{equation}
\left[\frac{d^{2}H}{d\xi^{2}}-2\xi\frac{d H}{d \xi }+...
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Calculating the average kinetic energy (expectation value) of gas particles from the Maxwell Boltzmann distribution
From what I already know, to calculate the expectation value/average from a probability distribution, you use the formula:
$$ \langle x \rangle \ = \int_{-\infty}^{\infty} x f(x) \,\mathrm{d}x \tag{1}$...
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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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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 ...
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