What Does $$E^2_k|_{k=k_{res}}$$ Mean?

In summary: In other words, the value of ##E^2_k## when k is equal to the resonance value, ##k_{res}##. This is used in the context of synchrotron radiation to calculate the energy density per unit wave vector.
  • #1
NODARman
57
13
Homework Statement
.
Relevant Equations
.
Hi, just wondering what this thing means.
$$
E^2_k|_{k=k_{res}}
$$
Just the k=k(res) after the vertical line. There is no definition in the textbook but in math does that mean from K=K(res) to something that can be dependent on a function or a situation?

Like definite integrals answer $$x|^3_2=3-2=1$$
 
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  • #2
NODARman said:
Homework Statement:: .
Relevant Equations:: .

Hi, just wondering what this thing means.
$$
E^2_k|_{k=k_{res}}
$$
Just the k=k(res) after the vertical line. There is no definition in the textbook but in math does that mean from K=K(res) to something that can be dependent on a function or a situation?

Like definite integrals answer $$x|^3_2=3-2=1$$
Without additional context it's hard to say. However, I don't think it's like a definite integral. Can you post a clear picture of the textbook page where this appears?
 
  • #3
$$
\left(\begin{array}{c}
D_{\psi \psi} \\
D_{\psi p}=D_{p \psi} \\
D_{p p}
\end{array}\right)=\left(\begin{array}{c}
\left.D \frac{\delta}{\gamma^2} E_k^2\right|_{k=k_{\text {res }}} \\
-\left.D \frac{\psi m c}{\gamma} E_k^2\right|_{k=k_{\text {res }}} \\
\left.D \frac{\psi^2 m^2 c^2}{\delta} E_k^2\right|_{k=k_{\text {res }}}
\end{array}\right),
\space where \space
E_k^2=\hbar \omega(k) n(k)=\int \frac{k^2 d \Omega}{(2 \pi)^2} \hbar \omega(\mathbf{k}) n(\mathbf{k})
$$
is energy density per unit of a one-dimensional wave vector and we assumed that ω(k) is an isotropic function of k.
we know that k is a wave vector (and the index "res" could be a doppler resonance for short) but what does it mean in that context (with E^2)?

This is from synchrotron radiation texbook.
Mark44 said:
Without additional context it's hard to say. However, I don't think it's like a definite integral. Can you post a clear picture of the textbook page where this appears?
I'll try to find the book.
 
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  • #4
NODARman said:
Hi, just wondering what this thing means.
$$
E^2_k|_{k=k_{res}}
$$
It means ##E^2_k## evaluated at ##k=k_{res}##.
 
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