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Questions tagged [hilbert-space]

Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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Law of addition in quantum physics

I don't know if the law of addition is broken in quantum physics. For example an electron is 100% wave and 100% particle, but one electron. is that 1+1=1. And do you get the same thing with ...
3
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2answers
266 views

Prove that if the expectation value of an operator in any state is 1, the operator is Identity

I want to prove that if $ \langle \psi | A | \psi \rangle = 1$ for all $ \psi ,$ then $A=I .$ Let's write $A$ and $\psi$ in the same basis. $$ \begin{alignat}{7} \left\langle \psi \middle| A \...
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0answers
27 views

Collapse of quantum state after measurement of degenerated eigenvalue (From textbook Shankar) (Closed)

I want to ask an easy question from Problem 4.2.1, Quantum Mechanics(2nd) by Shankar. Let's say Operators, $L_{x}$, $L_{y}$, $L_{z}$ are $$L_{x}$ = $1/2^{1/2}$ $\begin{pmatrix} 0& 1 &0 \\ ...
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3answers
110 views

Is there any intuitive reason behind why should the eigenfunctions of observables form a basis for our Hilbert space?

Is there any intuitive reason behind why should the eigenfunctions of observables form a basis for our Hilbert space ? For example, in the case of Stern-Gerlach experiment, sending the beam that has ...
1
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1answer
62 views

Why does superposition principle and Copenhagen interpretation not contradict with themselves?

In quantum mechanics, when we say that a particle in a state $|x_1\rangle$, physically the states $|x_1 \rangle $ and $c |x_1\rangle$ (for some $c\not = 0\in \mathbb{C}$) are the same, i.e they ...
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1answer
58 views

How do you expand a wavefunction in the basis of eigenfunctions of the free particle?

If we have an initial state given by $ \Psi(x,0) $ and we want to find $ \Psi(x,t) $, we would expand the function in the basis of eigenstates of the Hamiltonian, $\{\psi_n\}$: $ \Psi(x,t)=\sum _nC_n ...
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5answers
79 views

Confusion about ket states and bra with position

I am very confused about the bra-ket notation of states and the fact that $$\psi(x) = ⟨x|\psi⟩$$ and $$⟨x|x'⟩ = \delta(x-x')$$ are true. What does this mean? What is the ket $|x⟩$, is it just some ...
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0answers
45 views

Applying Sylvester's theorem in quantum mechanics

A $2d$ system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is an unitary Hermitian $2$x$2$ matrix with eigenvalues $\exp(±i\theta)$. I need to use ...
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0answers
23 views

Conceptual meaning of Thermal States

Thermal states are generally defined as $$\tau(\beta)= \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ What are some physical statements one can make about them? A system in thermal equilibrium is ...
3
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1answer
63 views

Tensor product of Hilbert space and Fock space

If we have system consist of two or more than two subsystems, we can write hilbert space of a system in terms of tensor product of Hilbert spaces of subsystems and we can write state of a system. But ...
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0answers
43 views

Calculating the evolution at any moment $t$ of a density matrix

I was reading the paper https://arxiv.org/abs/1303.4686, where we are given $N$ systems, all with the same Hamiltonian $$H=\sum_i \varepsilon_i \mid i\rangle\langle i\mid ~,$$ such that the joint ...
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0answers
37 views

Difference between pure and thermal states

As far as I know by inserting a harmonic potential $V(x) = \frac{1}{2}m \omega x^2$ into the time-independent schrödinger equation I can obtain the wave-functions eigenstates and eigenvalues (energies)...
0
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1answer
23 views

Is mode and orthonormal set of a particle the same thing? [closed]

Let's say we have a particle $A$ belongs to Hilbert space $H_A$. The complete orthonormal set of a particle is $\sum_{n=1}^{\infty}|\phi_n \rangle$ belonging to Hilbert space $H_A$. Now if we write ...
2
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1answer
66 views

Why do we use matrix product states?

Given a many body $\vert\psi\rangle$, we can express it in terms of a matrix product state. That is, $\vert\psi\rangle = \sum_{i,j..k}\psi_{i,j..k}\vert i,j..k\rangle$ can be rewritten as $\vert\...
5
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1answer
99 views

Dirac Delta Function and Position [duplicate]

How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
1
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1answer
74 views

distinguishable particles' Hamiltonian

Let us consider a classical Hamiltonian of a many body system \begin{equation*} H = \sum_{j=1}^N\frac{p_j^2}{2m}+V(\mathbf q) \end{equation*} and let us pass to quantum dynamics by promoting the ...
1
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0answers
21 views

How to measure off diagonal elements of mixed state density matrix?

Let’s assume I want to do quantum tomography using polarizers, half Waveplates and detectors , it’s obvious for me how we can measure diagonal elements of 2 qubit system density matrix using polarizer ...
1
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1answer
31 views

Action of rotation operator on spin 1/2 system

In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$ $$=e^{\...
2
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0answers
40 views

Expansion of an arbitrary density matrix in terms of coherent states?

It is well-known that any pure state can be expanded in terms of coherent states namely $$\left|\psi\right>=\frac{1}{\pi}\int d^2\alpha\left<\alpha|\psi\right>\left|\alpha\right>$$ due to ...
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1answer
32 views

What is the closure relation for multimode coherent state?

How does the closure relation for multimode coherent state $| \{ \alpha_\lambda \} \rangle $ look like? I suppose it should be some generalization of the closure relation for singlemode state $$\frac{...
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2answers
102 views

Representing tensor products using Dirac's bra-ket notation

I know, that $$ \uparrow \equiv \left[ \begin{array} { l } { 1 } \\ { 0 } \end{array} \right] $$ and $$ \bigg| \frac { X - i Y } { \sqrt { 2 } } \bigg \rangle = \sqrt { \frac { 3 } { 8 \pi } } \frac { ...
7
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1answer
156 views

Proof $\exp(-\beta H)$ trace-class operator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
1
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1answer
55 views

Computing a matrix element with the Wigner-Eckart-theorem

I learned about the Wigner-Eckart theorem and want to apply it to the following matrix element \begin{equation} \langle j \, m | r_kr_l | j' \, m'\rangle. \end{equation} I know this can be done by ...
0
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2answers
70 views

Change of basis from $J_y$'s eigenbasis to $J_z$'s eigenbasis for arbitrary $j$

I have been trying to compute the inner product ($j$ being fixed) $$\langle m_y|a_z\rangle \tag{1}$$ where $J_y|m_y\rangle = m |m_y\rangle$ and $J_z|a_z\rangle = a |a_z\rangle.$ I tried writing $$|m_y\...
2
votes
3answers
112 views

Operators commutation and relation between eigenvalues

If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
0
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2answers
72 views

How is the wave function Lebesgue integrable?

Let's assume we have a plane wave $\psi(x,t)= A_{0}e^{i(kx-wt)}$ in position space. To find the momentum representation of this wave we'd apply the Fourier transform. However, I don't see how this is ...
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2answers
88 views

Quadratic formula not working in Quantum Mechanics? [closed]

In quantum mechanics the raising operator of a system with quantum number $s$ and $m$ is such that $$\hat{S}^+|s,m\rangle = \hbar \sqrt{s(s+1)-m(m+1)}|s,m+1\rangle$$ Since there must exists a $m_\...
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2answers
72 views

Directional Eigenket?

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it. Because the angular dependence is ...
2
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0answers
26 views

Time dependence in Fock state representation

Say I was shooting a stream of single photons with a time interval between photons as $\Delta t$. Each time a photon is emitted, it is generated in a mode $\mathbf{k}_i$, where $i$ denotes the photon ...
7
votes
4answers
493 views

What is the difference between a Hilbert space of state vectors and a Hilbert space of square integrable wave functions?

I'm taking a course on quantum mechanics and I'm getting to the part where some of the mathematical foundations are being formulated more rigorously. However when it comes to Hilbert spaces, I'm ...
0
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1answer
52 views

Constructing a Hamiltonian for $N$-qubits

Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$. Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like? I ...
0
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1answer
46 views

Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate]

In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by $$I = \sum_i |x_i\rangle\langle x_i|.$$ To me, ...
8
votes
3answers
111 views

Does a physical interpretation of density matrix cross-terms exist?

Say we have some state $$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$$ it is in a quantum superposition of $|0\rangle$ and $|1\rangle$. Its density matrix is $$\rho=\begin{pmatrix}\frac 1 2 &...
1
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0answers
63 views

Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]

I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions". Given a finite dimensional vector space with a basis $\{x_i\}$, I understand ...
0
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0answers
77 views

Why are systems joined via a tensor product? [duplicate]

This question comes from seeing that the triangle addition rule for quantum mechanics comes out of groups/representation theory; I thought this was odd as we haven't used any group ideas in QM up to ...
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0answers
24 views

Wave equation IVP on arbitrarily shaped 2D domain

I’m interested in simulating a wavetable of arbitrary shape. Formally, suppose $R ⊂ \mathbb{R}^2$ is a region whose boundary is a simple closed curve $γ$. Let $u = 0$ on $γ$, and let $u$ satisfy ...
2
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2answers
71 views

Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization

It is well known that in elementary QM the so-called destruction/creation operators $$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$ are introduced when ...
1
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1answer
37 views

Completely positive maps - dimension of the ancilla space

If a map between positive operators $\Phi: X \rightarrow Y$ is also completely positive, it is true that $\Phi\otimes I_A$ is also a positive map for any choice of ancilla operator space $A$. That ...
0
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2answers
59 views

Linear combination of 2 spherical harmonic functions

The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?
0
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1answer
41 views

What do I get by multiplying a 0 operator on a 0 eigenvector?

I don't know how to write the equation form. Assuming my notation as Dirac notation, what do I get from $$ ( 0 | 0 | 0 ) ~?$$
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1answer
69 views

What is the content of an occupied QFT fermionic state?

A simple non-interacting quantum field is constructed by analogy to a harmonic oscillator, with $\hat{x}$ & $\hat{p}$ replaced by $\hat{ \phi}$ & $\hat{\pi}$ & with a separate oscillator ...
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0answers
26 views

Griffiths quantum mechanics evidence for a square-integrable solution must decay faster than $1/\sqrt{x}$ [duplicate]

In chapter 1.4 Normalization of Griffiths QM, the footnote of quote "Physically realizable states correspond to the square-integrable solutions to Schrodinger's equation" states that, Evidently $$ \...
3
votes
3answers
351 views

What is the “lowest energy”?

In many textbooks I come across the term lowest energy. For example in atomic structures, electrons are placed in orbitals in order for the atom to have the lowest energy. But what is this energy? ...
6
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3answers
123 views

Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
0
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2answers
36 views

Probability to get an Eigenvalue of Angular Momentum Operator on an Arbitrary Ket

Hello physics SE community, I am currently working on Principles of Quantum Mechanics by Shankar and i get stuck in page 336 (its not even an exercise). It basically said that "we may expand any $\...
18
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6answers
4k views

How do we know that entanglement allows measurement to instantly change the other particle's state? [duplicate]

I have never found experimental evidence that measuring one entangled particle causes the state of the other entangled particle to change, rather than just being revealed. Using the spin up spin down ...
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1answer
55 views

How does this term $e^{i\Phi_0}$ get removed in bloch sphere equation?

A qubit can be represented in the form of $$|\psi\rangle=\alpha|0⟩+\beta|1\rangle$$ where $\alpha$ and $\beta$ are complex numbers. Or a complex number can be expressed by $R e^{i\Phi_0}$. so the ...
4
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1answer
74 views

Physical significance of no self-adjoint momentum operator on half line?

I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...
2
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1answer
47 views

Why can we not define asymptotic states in CFTs?

I have known that we can't define asymptotic states in CFTs, because we can't use Fock spaces to describe CFTs. But is that right and why? I want to know some details about it.
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2answers
51 views

Proving the raising and lowering of the raising and lowering operator

I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$. However, in the written proof for $\hat A^{\...