Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to
$$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$
My question is how the above mentioned terms can be written as...
(Simplified version of Baym, Chapter 19, Problem 2)
Calculate, to first order in the inter-particle interaction V(r-r'), the energy of an N+1 particle system of spin-1/2 fermions with on particle of momentum p outside an N-particle Fermi sea (quasiparticle state). The answer should be expressed...
Homework Statement
After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin.
2. The attempt at a solution
I tried to apply the...
Homework Statement
Following from \hat{b}^\dagger_j\hat{b}_j(\hat{b}_j
\mid \Psi \rangle
)=(|B_-^j|^2-1)\hat{b}_j
\mid \Psi \rangle
, I want to prove that if I keep applying ##\hat{b}_j##, ## n_j##times, I'll get: (|B_-^j|^2-n_j)\hat{b}_j\hat{b}_j\hat{b}_j ...
\mid \Psi \rangle
.
Homework...
I'm studying plasmons from "Haken-Quantum Field Theory of Solids", and i need some help in the calculation of the equation of motion of eletrons' density
\begin{equation}
\hat{\rho}_{\overrightarrow{q}} = \frac{1}{\sqrt{V}} \sum_{\overrightarrow{k}}...
Homework Statement
I define the gamma matrices in this following representation:
\begin{align*}
\gamma^{0}=\begin{pmatrix}
\,\,0 & \mathbb{1}_{2}\,\,\\
\,\,\mathbb{1}_{2} & 0\,\,
\end{pmatrix},\qquad \gamma^{i}=\begin{pmatrix}
\,\,0 &\sigma^{i}\,\,\\
\,\,-\sigma^{i}...
Hello,
I've noticed that some professors will jump between second quantized creation/annihilation operators and wavefunctions rather easily. For instance \Psi_k \propto c_k + ac_k^{\dagger} with "a" some constant (complex possibly). I'm fairly familiar with the second quantized notation, and...
Homework Statement
Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators.
Homework Equations
Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
What the title says. Acting on a fermionic state with the number operator to a power is like acting with the fermionic operator itself. Does this allow us to define ## \hat{n}^k=\hat{n} ##? Or is there any picky mathematical reason not to do so?
I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field.
The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...
Hello,
I'm a bit confused about the eigenvalues of the second quantized fermionic field operators \psi(x)_a. Since these operators satisfy the condition \{\psi(x)_a, \psi(y)_b\} = 0 the eigenvalues should also anti-commute? Does this mean that the eigenvalues of \psi(x)_a are...
Hamiltonian of tight binding model in second quantization is given as H = -t \sum_{<i,j>} a_i^{\dagger} a_j
After changing basis it is H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}}
where E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}}
where \vec{b} is a nearest...
On page 9 of *Quantum theory of many-particle systems* by Alexander L. Fetter and John Dirk Walecka, during the derivation of the second-quantised kinetic term, there is an equality equation below:
>\begin{align}
\sum_{k=1}^{N} \sum_{W} & \langle E_k|T|W\rangle C(E_1, ..., E_{k-1}, W...
Hi.
I'll be doing a master's degree in nanophysics and working on electron transport in arrays of qubits.
I don't know anything (or barely) about the second quantization and would like a book which covers it, and on condensed matter overall.
So far I've been told about Bruus&Flensberg's...
Apparently, there are two different routes to get to quantum field theory from single-particle quantum mechanics: (I'm going to use nonrelativistic quantum mechanics for this discussion. I think the same issues apply in relativistic quantum mechanics.)
Route 1: Many-particle quantum mechanics...