I Understanding the Dirac Delta Identity to Fetter and Walecka's Formula

thatboi
Messages
130
Reaction score
20
Hi all,
I'm trying to verify the following formula (from Fetter and Walecka, just below equation (12.38)) but it doesn't quite make sense to me:
1698982648509.png

where
1698982665862.png
and
1698982691234.png

The authors are using the fact that ##\delta(ax) = |a|^{-1}\delta(x)## but to me, it seems like the ##\textbf{q}\cdot\textbf{k}-\frac{1}{2}q^{2}## are missing factors of ##\frac{1}{k_{F}^2}## right?
 
Physics news on Phys.org
From the definition you copied you get
$$q_0=\frac{\hbar k_F^2 \nu}{m}$$ and thus
$$\delta(q_0-\omega_{\boldsymbol{qk}})=\delta \left (\frac{\hbar k_F^2}{m} \nu -\frac{\hbar}{m} (\boldsymbol{q} \cdot \boldsymbol{k} + \frac{1}{2} q^2) \right)=\frac{m}{\hbar k_F^2} \delta \left (\nu- \frac{\boldsymbol{q} \cdot \boldsymbol{k} + q^2/2}{k_F^2} \right).$$
 
vanhees71 said:
From the definition you copied you get
$$q_0=\frac{\hbar k_F^2 \nu}{m}$$ and thus
$$\delta(q_0-\omega_{\boldsymbol{qk}})=\delta \left (\frac{\hbar k_F^2}{m} \nu -\frac{\hbar}{m} (\boldsymbol{q} \cdot \boldsymbol{k} + \frac{1}{2} q^2) \right)=\frac{m}{\hbar k_F^2} \delta \left (\nu- \frac{\boldsymbol{q} \cdot \boldsymbol{k} + q^2/2}{k_F^2} \right).$$
Ok cool that was what I got as well. Perhaps I missed some other definition along the way.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top