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1answer
66 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
32 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
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1answer
72 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
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2answers
113 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
59 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
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2answers
927 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
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1answer
26 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
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0answers
23 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
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1answer
39 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
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1answer
63 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
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1answer
30 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
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2answers
104 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
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1answer
30 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
61 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
59 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
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1answer
106 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
114 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
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0answers
46 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
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1answer
85 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
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1answer
113 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
121 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
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0answers
137 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
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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
62 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
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1answer
115 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
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0answers
48 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
269 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
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0answers
34 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
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1answer
44 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
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0answers
28 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
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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
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2answers
233 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
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0answers
454 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
144 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
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2answers
142 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
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1answer
91 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
130 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
175 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
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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
194 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 ...
1
vote
0answers
60 views

Why is this equation regarding boson coherent states true?

I'm reading the proof of the closure for boson coherent states and it involves the following step: $$ \int \prod_{\beta}\frac{\mathrm d \phi^*_{\beta} \mathrm d \phi_{\beta}}{2 \pi i} e^{-\sum_\beta ...
1
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1answer
115 views

How does vacuum state look in first quantization?

Wikipedia says that the vacuum state is the unit of tensor product. In my understanding then, a first-quantized wavefunction for the vacuum state would be just constant in the each particle's ...
0
votes
1answer
92 views

A problem about solving energy bands by the method of second quantization

In hopping model, we can get the Hamitonian as $H_0=-t\sum a^\dagger_ia_{i'}$. Then we take the fourier transform and put the operator which are in momentum space in the Hamitonian above. However, I ...
1
vote
1answer
92 views

Annihilation Operator on the Fock space

I agree that $$\hat a|0\rangle=0$$ But then, based on the above, the following should hold $$\hat a_k |N_1,...,N_{k-1},0,N_{k+1},...\rangle=|N_1\rangle\oplus\cdots\oplus |N_{k-1}\rangle\oplus \hat ...
0
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3answers
151 views

What is the general theory that describes the interactions between strings?

What is the general theory that describes the interactions between strings? I mean the basic object in the theory is (closed) string and they have interactions among them. The string theory, as I ...
6
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4answers
331 views

Why do we need $2^\text{nd}$ quantization of the Dirac equation

As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion, about it. So let me start with its definition: The Dirac equation, is given by $ i ...
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1answer
94 views

Hermiticity of the quantum field

The quantum field resultant from the quantization of a real classical field is hermitian, but why the quantum field corresponding to a complex classical field should be non-hermitian?
3
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1answer
113 views

Are constant terms in second-quantization relevant?

I have a rather broad question and a specific problem. Let's take a orthonormal single-particle basis $\{ \vert i \rangle \}$, a simple single-particle Hamiltonian $$\tilde{H} = \sum_{i, j} h_{i j} ...
1
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1answer
104 views

Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
11
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3answers
422 views

In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...