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4
votes
0answers
118 views

Has Sen quantized superstring fields?

Today I saw a paper by Ashoke Sen titled "BV Master Action for Heterotic and Type II String Field Theories". Is it really the "quantization" of superstring fields for the first time? What can be its ...
0
votes
2answers
49 views

Boson ladder operator $n+1$ factor [closed]

Looking at Boson creation and annihilation operators, I come across that \begin{equation} b_a|n_\alpha\rangle=\sqrt{n_\alpha}|n_\alpha-1\rangle \end{equation} and \begin{equation} ...
0
votes
1answer
29 views

One-electron reduced density matrix: Argument for positive semidefiniteness

I cannot follow an argument for positive-semidefiniteness of the one-electron density matrix given in "Molecular Electronic-Structure Theory" by Helgaker/Jorgensen/Olsen. First some definitions: ...
4
votes
1answer
53 views

Non-hermiticity of Dirac Lagrangian: null momentum?

The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are $$ \pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\ \bar \pi=\frac{\partial ...
1
vote
0answers
69 views

In QFT do we always use normal-ordered Hamiltonian? [duplicate]

In quantization of the Dirac field I learned that we use normal ordering to get rid of negative energy vacuum state. From this point is there any reason we would use original Hamiltonian?
2
votes
1answer
51 views

What is a single-phonon?

From what I understood from wikipedia, as well as some other resources, each phonon corresponds to a normal mode oscillation, and the creation operator to create a phonon of wavevector $k$ is: $$ ...
2
votes
1answer
44 views

How is intensity defined for quantized EM fields?

Classically intensity is defined as $$ I \equiv \frac{1}{2} c \epsilon_0 E^2, $$ but when you perform a second quantization this definition becomes a bit ambiguous since the $E^2$ could be ...
1
vote
3answers
140 views

Is Hamiltonian a differential operator in second quantization?

Normally, a free particle Hamiltonian is written $$ \hat{H} = - \frac{\hbar^2}{2m} \Delta $$ which is a differential operator because Laplacian $\Delta$ is. On the other hand, in second ...
0
votes
1answer
72 views

Commutation relations in second quantization

I know that for operators $a(\chi_1), a(\chi_2)$ of the same type (fermionic or bosonic) $$ [a(\chi_1), a(\chi_2)]_{-\xi} = [a^\dagger (\chi_1), a^\dagger (\chi_2)]_{-\xi} = 0 \tag{1}$$ where $$\xi ...
2
votes
1answer
56 views

Minus sign in the time ordering operator

The time ordering operator is usually defined as $$\mathcal{T} \left\{A(\tau) B(\tau')\right\} := \begin{cases} A(\tau) B(\tau') & \text{if } \tau > \tau', \\ \pm B(\tau')A(\tau) & \text{if ...
0
votes
1answer
69 views

second quantization lost particle number information?

In first quantization, we can read the particle number from the Hamiltonian. $$ H=\sum_{i=1}^N \hat{T}(x_i) $$ Converting this to second quantization form, the particle number of the system is lost: ...
0
votes
1answer
44 views

What is the form of the kinetic energy operator on a one dimensional (real space) lattice? (In second quantization)

I'm trying to figure out how one would write down the hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...
2
votes
1answer
75 views

Why are photons bosonic?

I am studying the quantization of the electromagnetic field. My text quantizes by changing amplitudes to ladder operators, by putting in an action and by imposing bosonic commutation relations upon ...
3
votes
2answers
135 views

Spin zero photons

As I understand it, the reason why there is no Spin 0 Photon is because the polarisation of an EM field lives in two dimension. Hence we only have two basis vectors, yielding two pairs of ladder ...
3
votes
1answer
64 views

Where can I find a detailed derivation of the form of two body operators in the second quantization?

I've been looking around online for a couple hours now and I can't find a very informative derivation of the form for two body operators in the second quantization. Is there a resource online ...
14
votes
2answers
994 views

What is the physical interpretation of second quantization?

One way that second quantization is motivated in an introductory text (QFT, Schwartz) is: The general solution to a Lorentz-invariant field equation is an integral over plane waves (Fourier ...
1
vote
1answer
30 views

Fourier transform of a set of L fermions operators

I have a set of L fermion creation and annihilation operators: $\lbrace{\hat{C}^+_1,...,\hat{C}^+_L\rbrace}$ and $\lbrace{\hat{C}^-_1,...,\hat{C}^-_L\rbrace}$. Every $\hat{C}^+_l,\hat{C}^-_l$ ...
1
vote
0answers
24 views

Creation and annihilation form of hamiltonian to derive a relation between the ac current applied to the crystal and the oscillations of the crystal

in the book "many-particle physics" by G.Mahan in piezoelectric subsection, it uses the second quantization formalism to derive the relation for hamiltonian of the electron-phonon interaction. so ...
1
vote
1answer
42 views

Multiply creation operator by a phase factor

A basic question, but I'm not completely confident what I'm doing is legit. I can multiply a creation operator by an arbitrary phase factor and it doesn't change any physics. True? I have a ...
0
votes
1answer
65 views

Fourier transform of random variables

My question is concerning Fourier transforms of random variables. So if the question itself is too heavy a task but you know of any good resources to learn this topic that would also be very much ...
0
votes
1answer
32 views

Why does trying to remove a non-existing electron from a state give zero?

Setup Creating an electron that is already in a basis set is zero (Pauli's principle): \begin{equation} a_i^+ | \chi_i \cdots \chi_k \cdots \chi_l \rangle = | \chi_i \chi_i \cdots \chi_k \cdots ...
1
vote
2answers
105 views

Two particle operator

Why is the two-particle (fermionic, cause for bosonic operators it is immediately clear that both representations are the same) Hamiltonian given by $$ H = \sum_{a,b,c,d} \langle ab|V|cd \rangle ...
2
votes
1answer
35 views

Slater-Determinant: When is this appopriate?

Imagine we have a N-particle Hamiltonian without any interaction between the electron particles $$ H = \sum_{i=1}^{N} \frac{p_i^2}{2m} + V(r_i)$$ then the solution to this equation $H\Psi = E \Psi$ ...
0
votes
2answers
62 views

Schrodinger field and klein gordon field

In the usual Fourier expansion of schrodinger fields \begin{align} \Psi(\vec{x}) = \frac{1}{(2\pi)^{\frac{3}{2}}} \int d^3 k \hat{a}_k e^{-i (wt-\vec{k}\cdot \vec{x})}, \quad \Psi^{*}(\vec{x}) = ...
1
vote
1answer
61 views

Product of deltas in kinetic second quantization hamiltonian

I am trying to derive the result for a kinetic hamiltonian in second quantization in term of the fields, that is: $\hat{H} = \int - \Psi^\dagger (r) \frac{\hbar^2\hat{\nabla}^2}{2m} \Psi(r)$ I start ...
0
votes
1answer
119 views

Is there something wrong with quantizing two times in second quantization?

Second quantization is sometimes considered to be a bad name, because a single quantization is enough. For electrons, we can either start from a many body viewpoint and introduce field operators or we ...
0
votes
1answer
119 views

Perspectives of QFT [closed]

From the answer to this question Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, I have discovered that there is two perspectives to QFT. I am doing a course which is unfortunately a summary of QFT and ...
0
votes
0answers
50 views

Hamiltonian for semiconductor

I was wondering which terms we need in a semiconductor Hamiltonian where no transition between the valence and conduction band occur? First we would have a term describing the energy of the full ...
0
votes
1answer
91 views

Time-dependence of ladder operators in quantized EM fields

My Question Are the operators for the $A$, $E$ and $B$ field to be treated as operators in a Heisenberg description or is their time dependence explicit when performing a textbook EM quantization as ...
1
vote
1answer
115 views

Second Quantisation, Fourier Transform, minus sign [closed]

I want to expand a field \begin{equation} \Phi (x) = \int \frac{d^3 p}{(2 \pi)^3} e^{ipx} \end{equation} in terms of the second quantisation \begin{equation} \Phi = \frac{1}{\sqrt{2 E}} (a + ...
0
votes
1answer
125 views

Contructive Proof of 2nd Quantization form of Operators

Is there a constructive proof for these forms of operators in second quantization $$R= \sum \limits_a \sum \limits_b \langle a | R_1 | b \rangle C_a^\dagger C_b $$ using the general form $R = \sum ...
1
vote
0answers
163 views

Derivation of Rashba spin-orbit coupling in tight-binding model

Rashba spin-orbit coupling Hamiltonian in free space can be written as: $H_{\text{so}}=\int d^3r \Psi^{\dagger}(\mathbf{r}) \gamma (p_{x}\sigma _{y}-p_{y}\sigma _{x})\Psi(\mathbf{r})$. I expand ...
0
votes
0answers
42 views

What is the missing step in this result regarding the creation operators in Fock space?

In the above extract from Simons and Altman: Condensed Matter Field Theory, I am having trouble getting from (2.3) to (2.4) in the case of Fermions (ζ=-1 and the n(subscript i) values are modulo 2). ...
0
votes
1answer
63 views

Inverting the field creation operator $|\Psi\rangle$

In my lecture notes on second quantization it is written that the creation field operator is given by $|\Psi\rangle^{\dagger}_s (r) = \frac{1}{\sqrt{V}} \sum_{k} e^{-i k r} \hat{a}^{\dagger}_{ks}$ ...
1
vote
1answer
129 views

numerical diagonalization of tight-binding hamiltonian

I would like to find the exact eigenvalues of the following tight-binding Hamiltonian, written here in second quatization: \begin{eqnarray} \hspace{-0.25in}{\mathcal{H}} &=& \mathcal{H}_0+ ...
0
votes
0answers
54 views

piezoelectric in quartz

Does any one know if it is possible to find the relation between the ac current frequency applied to a piezoelectric and the change in the crystal lattice due to this current BY USE OF HAMILTONIAN (in ...
4
votes
4answers
273 views

What is the right order of creation operators?

I started to learn some basics of second quantisation and specifically its use in quantum chemistry. Currently I'm reading this book by Péter R. Surján, and here is small excerpt from it. If one ...
0
votes
0answers
42 views

Linear Canonical Transformation in Berezin's book on Second Quantization

This question pertains to linear canonical transformations for bosons in chapter II of Berezin's book "The Method of Second Quantization". Berezin considers a linear transformation of creation and ...
1
vote
1answer
59 views

Hamiltonian for electron hole

I found in lectures notes that the Hamiltonian containing the energy of a electron hole without any interaction is given by $$H = \sum_k d_k^{\dagger} d_k \left( \frac{\hbar k^2}{2m_V} - E_{0,V} ...
0
votes
0answers
29 views

Two state Hubbard modell

I am given the two state Hamiltonian $$ H = U \sum_{j \in \{L,R\}} n_{j \uparrow}n_{j \downarrow} - t \sum_{\sigma \in \{\uparrow,\downarrow\}}(a_{L \sigma}^{\dagger}a_{R \sigma} +a_{R ...
0
votes
0answers
18 views

States in valence and conduction band

I often see a Hamiltonian in second quantization written for the valence and conduction band. Now, I was wondering: What are the single-electron states that form the prouct state they act on? So what ...
0
votes
2answers
262 views

Creation and annihilation operators in Hamiltonian

If I find a Hamiltonian $H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_k V_k a_k^{\dagger} a_k$ then I was wondering: As far as I know this is many body theory and so these operators act on ...
8
votes
0answers
470 views

Horrifying electron gas model

I am given the Hamiltonian, in an exercise called plasmons, and where $\langle, \rangle $ denotes the expectation value. $$ H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_{k_1,k_2,q} V_q ...
3
votes
2answers
159 views

What is the physical interpretation of a field operator

So far in our lecture we defined creation operators $a^{\dagger}_{n}$ in the following way, that we said: Somebody got you a antisymmetric or symmetric N- particle state and now $a^{\dagger}_{n}$ ...
3
votes
2answers
158 views

Describing a single photon with creation and annihilation operators

Since I am not fully aware of the creation and annihilation operator formalism for single photons, I want to ask, if the following is correct: I am considering a photon in the vacuum which travel ...
1
vote
1answer
92 views

Creation and annihilation operators

In our lecture today, we introduced two kinds of creation and annihilation operators. I want to restrict myself to the antisymmetric case: The first operator $a_k^{\dagger}$ creates a state ...
0
votes
2answers
134 views

Second quantization, creation and annihilation operators

I found two notions of states for second quantization. One representation uses occupation numbers here, for example Another one creates the n+1 th particle in a collection of n existent states. see ...
2
votes
1answer
216 views

Time reversal operator in tight-binding model with second quantization form

In the tight binding model, $H=\sum_{r,r'}ta^{\dagger}_{r}a_{r'}+h.c.$. When conducting a time reversal transformation, what form will this Hamiltonian take? Or how can I express time reversal ...
3
votes
1answer
71 views

Bosonic Schrödinger field [closed]

When second quantizating the Schrödinger field $$\psi(r,t) = \sum_i \phi_i(r)b_i(t),\quad\mbox{and}\quad \psi^{\dagger}(r,t) = \sum_i \phi_{i}(r)^* b_i^{\dagger}(t),$$ we have the commutation ...
3
votes
2answers
212 views

Normal Ordering the $\phi^4$ interaction

I am trying to quantize the quartic potential $(\lambda/4!)\phi^{4}$ in a box of side length $L$, with periodic boundary conditions. I have expanded the field $$\phi = \sum \limits_{\vec{n}} \exp(i ...