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

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42 views

Transforming to a rotating frame in the $x$-basis

I was reading this paper on analytically Solvable driven time-dependent two level quantum systems. The Hamiltonian considered in the paper is the following: $$H=\sigma_z\cdot J(t)/2)+\sigma_x\cdot ...
0
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0answers
23 views

Show that partial derivative with respect to time is anti-hermitian [on hold]

I have a definition that $$<s_1,s_2> = \int_{-\infty}^{+\infty}s_1^*(t)s_2(t) dt$$ I need to show that $\partial_t$ operator which is just the partial derivative with respect to time is ...
2
votes
1answer
51 views

Decoupling coupled differential equations in dynamically coupled two state system

Consider the following dynamically coupled two state hamiltonian, $$H=-B\sigma_z-V(t)\sigma_x.$$Taking the eigenfunctions of $\sigma_z$ ($|+>$ and $|- >$) as basis vectors, we have the wave ...
-2
votes
0answers
32 views

Why do we use a vector space for gluons? [on hold]

If the theory cannot be approximated as an harmonic oscillator I don't see why quantized $e^{ikx}$ mode would make sense as a description of gluons. None of these modes individually solves the Yang ...
3
votes
5answers
352 views

Physical interpretation of applying a unitary operator to a state

When we apply one of the Pauli matrices $\sigma_y$ on one of its eigen-vectors $| \odot \rangle$, what does the eigen-value tell us about $| \odot \rangle$? Is this considered a measurement of $| ...
1
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1answer
22 views

Annihilation operator in harmonic oscillator

In Wikipedia's QHO page there is a moment when the following is stated: I don't know why "the ground state in the position representation is determined by $a|0\rangle=0$". I would say that the ...
0
votes
2answers
40 views

Transformation of $| JM\rangle$ under the group of rotations

I am following the Quantum Mechanics I, Galindo A., Pascual P. and in page 207 explaining the matrix representations of the Rotation Operators in the angular momentum it appears the next (obvious) ...
5
votes
1answer
73 views

Heisenberg's uncertainty principle derivation in a ring [duplicate]

The standard derivation But now suppose the space is a ring of length $L$, it seems the derivation could work out exactly the same and we get $$\Delta p \Delta x \geq \hbar/2.$$ But since $\Delta x$ ...
-4
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1answer
48 views

Eigenvalues, Eigenvectors [on hold]

what is the physical meaning of eigenvalues and eigenvectors? Is there any relation between them and the energy states if there is any reference please provide it
15
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1answer
1k views

Is the existence of a sole particle in an hypothetical infinite empty space explicitly forbidden by QM?

Suppose the universe is completely empty with one sole particle trapped in it. To simplify, I will only be looking at the one dimensional case. However, all arguments are applicable for three ...
7
votes
2answers
1k views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
4
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1answer
148 views

BRST quantization and norm

States with definite ghost number have zero norm (since ghost number is anti-hermitian and has real eigenvalues). E.G. when quantizing relativistic point particle, physical spectrum turns out to ...
3
votes
1answer
42 views

Constructing a POVM to discriminate $m$ quantum states. What if they're linearly dependent?

I've come across this problem in Nielsen & Chuang's Quantum Information book (problem 2.64) Suppose Bob is given a quantum state chosen from a set $|ψ_1 \rangle, . . . , |ψ_m\rangle$ of linearly ...
2
votes
1answer
48 views

Using symmetry to determine a hydrogen electron's decay route from $|300\rangle$ to $|100\rangle$

Lets say we have an electron in state $|nlm\rangle = |300\rangle$ of the hydrogen atom. By selection rules, we know that it can only decay to ground state in 3 ways, namely through the $|21m\rangle$ ...
3
votes
1answer
186 views

Density matrix: error with diagonalization claim and fixing it

On page 174 of Townsend's "A Modern Approach to Quantum Mechanics", 2nd edition, it says the following: "For a mixed state, one for which $p_k$ is the probability that a particle is in the state ...
1
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1answer
56 views

Time-ordered product of two normal-ordered products of fields

Suppose you have a scalar field theory with field operators $\phi(x)=\phi(x)_+ + \phi(x)_- $ that can be decomposed into terms of annihilation and destruction operators. Let $$ D(x-y) = ...
0
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1answer
53 views

Quantum Mechanics: Rotation operators

How do I know what direction of the rotation operator to use on the initial state of a spin-1/2 particle? For example, a spin-1/2 particle initially in the $\lvert y \rangle$ state enters a SGz ...
2
votes
1answer
64 views

Diagonalisation: Schmidt vs eigenvalue - when to use which?

In physics we encounter diagonalisation of matrices or operators in a variety of areas. But there are different kinds, the main two being Schmidt decomposition and eigenvalue diagonalisation. The two ...
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0answers
31 views

Intuitive understanding of vector spaces [closed]

I want to understand the intuitive meaning of vector spaces (Hilbert, Banach, metric, normed). I have read a lot of texts and understand the mathematical formalism given but how does it correspond to ...
12
votes
3answers
5k views

Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
0
votes
2answers
44 views

Order of operators and numbers inside a bracket

I had an argument with my professor. Let $H$ be an operator (e.g. hamiltonian). Let capital $X$ denote the position operator. Let $f$ and $g$ be functions of $X$ that do NOT commute with $H$. Now ...
0
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1answer
62 views
-5
votes
1answer
41 views

CNOT Gate for quantum systems [closed]

A you know the CNOT gate is 4 by 4 matrix, is there any way to show it by a 2*2 matrix? if yes what will be the elements?
3
votes
1answer
111 views

How do (and don’t) particles emerge from fields?

I am aware of the following field- and particle-like notions: QFT particle, a unit of excitation in (the Fock space of) a QFT; SR field, an extremal $A = A(\mathbf x)$ of a Lorentz-invariant action; ...
0
votes
1answer
29 views

Tensor products of Hilberts spaces: definition of outer products and commutators

Suppose one has two single-particle Hilbert spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ and consider the tensor product of these such that $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a two-particle ...
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1answer
281 views

Simple QFT simulation - how to do it

I would like to write a simple QFT simulation for a free scalar field with a cubic interaction term. However, I got stuck a bit. I will try to describe what I think I understand. I want to have a ...
3
votes
2answers
78 views

What is the qualitative difference between quantum superpostion and mixed states? [duplicate]

As I understand it, if one has a complete knowledge of the state of a quantum system (insofar as one knows the statistical distributions of all the observables associated with the state) then one can ...
1
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0answers
54 views

How did Max Born come up with his rule? [duplicate]

In his rule, he stated that the probability is norm-squared of wave function, $|\psi|^2$. And as far as I knew, no one else at that time had "right" interpretation of the wave function. Even ...
9
votes
2answers
272 views

What is meant by the term “completeness relation”

From my humble (physicist) mathematics training, I have a vague notion of what a Hilbert space actually is mathematically, i.e. an inner product space that is complete, with completeness in this sense ...
11
votes
5answers
254 views

How is the ground state chosen in a spontaneous symmetry breaking process?

This question is about how the ground state is chosen in a spontaneous symmetry breaking process. Say we have a Mexican Hat potential (e.g. the one for the Higgs field) and are sitting at the unstable ...
-3
votes
2answers
82 views

Is the Wave Function a Unitary Operator? [closed]

A unitary operator can be represented as an exponential $$e^{iA}$$ and as we represent the wave function in general as $$e^{i k x}.$$ Does that mean that the wavefunction is unitary as the exponent is ...
4
votes
2answers
105 views

Where the time-dependent wavefunction $\Psi(\vec{x},t)$ lies?

Supose $\vec{x}=(x,y,z)\in \mathbb{R}^3$. The state of a physical system is described by the function $\Psi(\vec{x},t)$, where it must satisfy $$\int_{\mathbb{R}^3} ...
4
votes
1answer
162 views

How to make rigorous the idea of a continuous complete set?

In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let $\mathcal{H}$ be ...
0
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1answer
53 views

Measurement on two Qubits

Assuming I have two Qubits, i.e. a four-dim. Hilbert space. In the following, I choose the basis {|11>,|10>,|01>,|00>}. I want to have a look on the non-diagonal part <11|$\rho$|00>. How can I ...
0
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0answers
76 views

Eigenfunctions of translation operator

I had an HW assignment in which we were asked to find the eigenfunctions of the translation operator which is defined as follows: $$\hat{D}(a)=e^{-(i/\hbar)a\hat{P}}$$ where $\hat{P}$ is the momentum ...
10
votes
1answer
311 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
-1
votes
2answers
445 views

Expectation values of commutator and anti-commutator (momentum and position)

What are the expectation values of commutator and anti-commutator for momentum and position operators? In the case of commutator: $$\langle[x,p]\rangle=\langle i\hbar\rangle=~?$$ In the case of ...
13
votes
4answers
2k views

What is a wave function in simple language?

In my textbook it is given that 'The wave function describes the position and state of the electron and its square gives the probability density of electrons.' Can someone give me a very ...
0
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0answers
10 views

The Wigner angle for two-particle state

Suppose we have the Wigner angle $\theta (\mathbf k, \Lambda)$, which is defined through the Lorentz group transformation $U(\Lambda)$ of one-particle state $|\mathbf k , \sigma\rangle$ ($\sigma$ ...
1
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0answers
34 views

How to represent the spherical wave by using Fock basis?

Suppose I have two particles with opposite momentum: $$ |\psi \rangle_{\mathbf k} = |\mathbf k; -\mathbf k\rangle ,\quad |\mathbf k| = M $$ I want to represent the spherical symmetric distribution of ...
0
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1answer
23 views

How to get explicit value of Wigner angle for massless one-particle state transformation?

The one-particle massless state $|\mathbf p , \sigma\rangle$ is transformed under the Lorentz group $U(\Lambda) \equiv U(\Lambda , 0)$ as $$ U(\Lambda)|\mathbf p, \sigma \rangle = \sqrt{\frac{(\Lambda ...
0
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0answers
23 views

Continuum of states after 2-particle states

In the Hilbert space of some free theory one can define single-particle states as $|\vec{p}>$, 2-particle states as $|\vec{p},\vec{q}>$ and so on. The $total$ 4-momentum eigenvalue of the ...
3
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1answer
38 views

Generators of a certain symmetry in Quantum Mechanics

In Classical Mechanics to describe symmetries like translations and rotations we use diffeomorphisms on the configuration manifold. In Quantum Mechanics we use unitary operators in state space. We ...
0
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0answers
42 views

Extending projection operator to infinite-dimensional case

Hi I have a basic question regarding bra-ket notation. Given that $\{|e_n \rangle \}$ is a discrete orthonormal basis, $$\langle e_m | e_n \rangle = \delta_{mn}$$ then $$\sum_{n}|e_n \rangle \langle ...
0
votes
1answer
68 views

Dirac Notation With Comma

Does $\langle A,B\rvert$ mean $\langle A\rvert\langle B\rvert$? If so how is an operator applied to this in $\langle A,B\rvert \hat O $? For an example say the annihilation operator acting on ...
11
votes
2answers
513 views

Tensor product in quantum mechanics?

I often see many-body systems in QM represented in terms of a tensor products of the individual wave functions. Like, given two wave functions with basis vectors $|A\rangle$ and $|B\rangle$, belonging ...
0
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1answer
44 views

Allowed Wave Functions of System

Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given ...
4
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3answers
102 views

Same quantum states represented in different basis

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose and then ...
3
votes
2answers
142 views

Schrödinger equation in momentum space

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose. It then ...
0
votes
1answer
125 views

Expectation value of total angular momentum $\langle J \rangle$

[I am working with Griffiths Introduction to Quantum Mechanics, 3rd Edition. My problem is general but if you want to look I am reading from ch 4.1 in which the weak-field Zeeman Effect is being ...