Questions tagged [hilbert-space]

This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete or closed. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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Does the abstract wavefunction change in this following example?

Suppose, we have a basis $|u\rangle$, described by the function $u=g(x)$. We can normalize this basis, using our standard $|x\rangle$ basis using the following : $$\hat{I}=\int |x\rangle\langle x|dx=\...
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Normalizable wavefunctions for bound states

In my quantum mechanics book I read the following sentence: If we want the wave function to be normalizable, one must impose boundary conditions: $$\lim_{x \to \pm\infty} \psi(x)=0.$$ My question is ...
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48 views

Dirac Delta function expressed in terms of molecular orbital basis set

In the book of Helgaker 1 on page 16 is written that for a complete one-electron molecular orbital basis, the Dirac delta function may be written in the form of: $$\delta(\mathbf{x} -\mathbf{x'})=\...
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1answer
59 views

Inserting a position operator in the path integral in QFT

With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t''...
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1answer
43 views

How do we choose the basis of an Hilbert Space?

When we define a basis for the Hilbert Space for a spin half particle I understand it being done using the principle of mutual exclusivity that is if $S_z = +\hbar/2$ then it cannot be $S_z = -\hbar/2$...
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1answer
47 views

Average squared of angular momentum

Consider spin $S$ particles. Is there an easy way to prove that $\langle \vec{S} \rangle \cdot \langle \vec{S} \rangle \leq S^2$ for all states? I understand intuitively that the bound is realized ...
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On the invariance of $ \rho |\psi \rangle$? (For Wigner and his friend)

Motivation So let's say I shoot circular polarized light which is either right $| R \rangle$ or left $| L \rangle$ polarized. My friend notices that I choose left and right equally $50$% of the time. ...
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755 views

What does mathematical consistency in QFT mean?

My question is more naive than Is QFT mathematically self-consistent? Just when people talk about the mathematical consistency of QFT, what does consistency mean? Do people want to fit QFT into ZFC? ...
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43 views

Do I have to change the $S_z$ eigenbasis to $S_x$ or $S_y$ eigenbasis to calculate statistical properties for $S_x$ or $S_y$?

I'd like to know what the quantum average values $\langle S_x \rangle$ and $\langle S_x \rangle$ are given the initial state of a system in the $S_z$ eigenbasis $|\psi(0)\rangle = N|1\rangle + 2i|2\...
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Selection Rules using Group Theory

I was learning about the applications of Group Theory and one important application is looking at the selection rules in a weak EM field. We essentially want to see whether the matrix element $\langle ...
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48 views

Does choice of renormalisation scheme affect the consequences of Haag's theorem?

So Haag's theorem means that the interaction and Hamiltonian picture are not equivalent. The reason seems to be that renormalization mixes interactions and free particles (ie self energy of a free ...
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Do states with infinite average energy make sense?

Do states with infinite average energy make sense? For the sake of concreteness consider a harmonic oscillator with the Hamiltonian $H=a^\dagger a$ and eigenstates $H|n\rangle=n|n\rangle$, $\langle n|...
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How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle $? [closed]

How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle $ ? $\varepsilon$ is a polarization vector. I also want to know the lists on the webpage that says some reduced matrix ...
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XOR/CNOT on 2 qubits in superposition [closed]

What is $(|0⟩ - |1⟩) ⊕ (|0⟩ + |1⟩) = ? $ or more elaborate what is the result of the CNOT gate when a and b are in superpositions CNOT$|a,b⟩→|a,a⊕b⟩$, having $a = (|0⟩ - |1⟩)$ and $b = (|0⟩ + |1⟩)$
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59 views

Proof that joint-measurability means commutativity

For $i=1,2$, two measurements $m_{i}:\mathcal{X}_{i}\to\mathcal{L}(\mathcal{H})$, from alphabet $\mathcal{X}_{i}$ to set of bounded linear operators on Hilbert space $\mathcal{H}$, are compatible or ...
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What is the dimension of two quantum systems with bases $\{a_1,a_2\}$ and $\{b_1,b_2,b_3\}$, combined? [closed]

Quantum system A has a basis $\{a_1, a_2\}$. System B has a basis $\{b_1, b_2, b_3\}$. A and B evolve according to their own Hamiltonian and do not interact at all. If I consider A and B as one large ...
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80 views

Derivation of Husimi $Q$-function of Squeezed vacuum state [closed]

I am trying to derive the expression for the Husimi $Q$-function of the squeezed vacuum state using the fock state representation of the squeezed vacuum state. $$|\mathrm{\zeta}\rangle=\frac{1}{\sqrt{\...
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6answers
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What is the actual use of Hilbert spaces in quantum mechanics?

I'm slowly learning the quirks of quantum mechanics. One thing tripping me up is... while (I think) I grasp the concept, most texts and sources speak of how Hilbert spaces/linear algebra are so useful ...
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1answer
79 views

Is there a well-defined association between abstract linear operators in Fock space and normal ordered polynomials of fermionic operators?

Suppose I have a fermionic Fock space $H$ of dimension $2^n$. If I fix an operator $O$ acting on $H$ that commutes with the number operator $N$, I typically make an assumption internally that such an ...
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Why are only positive frequency mode functions allowed in Quantum field theory? How is this consistent with anti particles having negative energy?

In quantum field theory, one can redefine the particle creation and annihilation operators using Bogoliobov transformations, which can give rise to a different vacuum state, using a new set of ...
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Expectation value of position operator for driven quantum harmonic oscillator

I'm having quantum harmonic oscillator which is in ground state at $t=0$. Now it is subjected to arbitrary force $F(t)$. How can I calculate $\langle \hat{x} \rangle$ and $\langle \hat{p} \rangle$? I ...
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1answer
56 views

Separability of Solutions of Schrödinger Equation

In my physics class we always solve the Schrödinger Equation using separation of variables. However, this makes me wonder if we are really getting all solutions to the Schrödinger Equations in this ...
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50 views

Are all physically realistic Hamiltonians local?

My understanding of modern physics is that physicists think that, fundamentally, physical laws are local. For system A to interact with system B, they either need to be very close to each other or ...
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17 views

Iterative block diagonalisation in degenerate perturbation theory

How is iterative block diagonalisation carried out in the matrix form of degenerate perturbation theory to lift degeneracy present in the zeroth-order or unperturbed Hamiltonian ($H_{0}$)? In general ...
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1answer
71 views

Meaning of Scattering Amplitude in QFT

I've been reading a textbook on QFT, and learned that we can calculate the probability amplitude that a system in state $|i\rangle$ will "collapse" into state $|f\rangle$ after some amount ...
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1answer
52 views

Angular momentum sum of two 3/2 spin particles

I have a question about the sum of angular momenta of two $3/2$ spin particles (considering no orbital angular momentum). Let's suppose that I can with a magnetic field collide two $3/2$ spin ...
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1answer
41 views

Mixed state in single mode Light?

Most quantum optical textbooks introduce thermal light (blackbody radiation) as an example of a mixed state. And those states covered in most of the textbook represents a Fock state, the state of the ...
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3answers
556 views

Why are two-electron systems usually described in singlet-triplet basis?

Why are the two-electron system usually described in singlet-triplet basis, but not computational basis $\uparrow\uparrow$,$\uparrow\downarrow$,$\downarrow\uparrow$,$\downarrow\downarrow$? What is the ...
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1answer
68 views

What is $\langle 0|p\rangle$?

$\hat{p}$ is the generator of the translation group, so $$|r\rangle=e^{-ir\hat{p}/\hbar}|0\rangle\to\langle p|r\rangle=e^{-irp/\hbar}\langle p|0\rangle.$$ Assuming normalized position states \begin{...
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56 views

Can the Hermitian operator be related to state space to describe physical phenomena?

Can space-time, in which phenomena occur, and the space of states in which phenomena are described by means of the Hermitian operator be related? I suspect it is because the hermetic operator is built ...
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Why $⟨a│H'│a⟩=0$, when $H'$ has zero diagonal components?

On 'David J.Griffiths, Introduction to Quantum Mechanics' Chapter 11.1.1 equations (11.16) When $H'$ has zero diagonal component, $⟨a│H'│a⟩=0$ However, I think that there is needed conditon '...
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1answer
73 views

Quark model wavefunction separability

When looking at the quark model we say we can write the wavefunction in 4 components, ie $$ |\Psi \rangle = |\Psi_{spatial} \rangle|\Psi_{spin} \rangle|\Psi_{flavour} \rangle|\Psi_{colour} \rangle \...
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76 views

Commutation between hermitian operator $H$ and the operator $U(m,n) = |\varphi_m\rangle \langle \varphi_n|$

I was trying to do this exercise from Cohen-Tannoudji Quantum Mechanics book: $|\varphi_n\rangle$ are the eigenstates of a Hermitian operator $H$ ( $H$ is, for example, the Hamiltonian of some ...
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61 views

What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?

While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is ...
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76 views

Do fermionic creation/annihilation anticommutation relations fix the creation and annihilation operators?

If you define operators $a, a^\dagger$ which satisfy (e.g.) the relations $\{a,a\}=\{a^\dagger,a^\dagger\}=0$ and $\{a,a^\dagger\}=1$. Will this uniquely define the operators such that $a |0\rangle \...
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1answer
43 views

Requirement of Jordan-Wigner string in creation operator on Fock state

Our lecture notes described the action of the particle creation operator on a fermionic Fock state: $$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$ I am ...
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1answer
21 views

Eigenstates of a spin 1 object [closed]

Since the eigenvalues of $S_z$ are -1, 0, 1, is it correct to say that the eigenvectors are $$\begin{bmatrix} 1\\0\\0\end{bmatrix},\begin{bmatrix} 0\\0\\0\end{bmatrix}, \textrm{and} \, \begin{bmatrix} ...
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1answer
61 views

$S_x$ and $S_y$ states for spin 1

I asked a question earlier but it looks like I misunderstood something Convert eigenvectors to different basis. I'm considering the case of a spin 1 object, where the eigenvalues of $S_z$ are 1,0,-1 ...
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43 views

Convert eigenvectors to different basis

For a spin 1 object, the eigenvalues of $S_z$ are 1,0,-1 so the $S_z$ diagonal basis is just $\{|1\rangle, |0\rangle, |-1 \rangle\}$ and the $S_z$ and $S_y$ operator are respectively $$S_z = \hbar \...
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1answer
64 views

Confusion about interpretation of expectation values in quantum mechanics

Given a state $|\psi \rangle$ one can form the expectation value of an observable $O$ as: $$ \langle \psi|O|\psi \rangle. $$ For the case $O = H$, where $H$ is the Hamiltonian of the quantum system, ...
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158 views

Confusion about in and out states, interacting Hilbert space etc, referring to Weinberg QFT

There are many posts related to this issue on this site, but I have found none that answer my specific questions about this matter. I review my understanding of Weinbergs approach. There are probably ...
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50 views

Adiabatic turn-on of free multi-particle states

Consider a second-quantized operator $\mathcal{H}_{full}$ describing some interacting QFT, whose action is known on a set of Fock states $\{\mathcal{|F\rangle}\}$, which, in turn, are the eigenstates ...
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Normalization of One-Particle States for Klein-Gordon Field Quantization

Peskin & Schroeder in their QFT textbook discusses how we may normalize one-particle states $|\textbf{p}\rangle$ for Klein-Gordon field quantization in pages 22-23. The excerpts are given below. ...
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1answer
76 views

What exactly is a Fock state?

I am a bit confused by the way a Fock state is defined and hope to find some clarification. The Fock space is defined as the direct sum of all $n$-particle Hilbertspaces $H_i$ $$F = H_0 \oplus H_1 \...
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2answers
81 views

Definition of single-particle states in the free theory

I like defining single-particle states as simultaneous eigenstates of generators of the Poincare group (basically, the representations of the Poincare group). This is the most fundamental definition ...
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1answer
44 views

Existence of Ground State of Dirac equation

In chapter four of Ryder, the author showed that there exists a ground state $|0\rangle$ for the Kelin-Gordon equation, just like the case of the linear harmonic oscillator. However, I was not able to ...
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1answer
80 views

Normalisation in Dirac Notation

Say I have a wave function as follows (example): $$|\psi\rangle=|\phi_1\rangle-\sqrt{3}|\phi_2\rangle+ 2i|\phi_3\rangle$$ I know normalisation means: $$\langle \Psi_N|\Psi_N \rangle =1\tag{1}$$ I know ...
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4answers
123 views

Shared eigenstates of angular momentum

The angular momentum operators $L_x$, $L_y$ and $L_z$ don't commute with each other but they do all commute with the operator $L^2 = L_x^2+L_y^2+L_z^2$. I know that if two matrices $A$ and $B$ commute ...
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How to calculate the kinetic energy when the state is not vacuum state?

This question is related to How to calculate the kinetic energy in field theory when the state is not vaccum state Suppose we have a state $|\psi\rangle=\int dx \exp\left(\alpha(x)\hat{a}^{2}(x)-\...
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1answer
48 views

Matrix representation of the operator $\vec{Z}: \langle l^{'} , m^{'} | \vec{Z} | l, m \rangle $

I know that $[L_z, z] = 0$, which means that the operators $L_z$ and $\vec{Z}$ share a common eigenspace. So, I wonder how I can get the matrix representation of the operator $\vec{Z}$ in the ...

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