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

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92 views

What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
2
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3answers
100 views

I am learning Quantum Mechanics and I have some questions about some basic concept [closed]

What does a "STATE" exactly mean in quantum mechanics? What is the equivalence of "STATE" in classical mechanics? If we have a wave function $\Psi$ , its absolute square $|\Psi|^2$ is the ...
2
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2answers
95 views

Bloch Sphere and $SU(2) \to SO(3)$ map

For any matrix $U \in SU(2)$ there is an associated map from $S^2$ (the surface of a 3-disk) to itself defined by $\pi \circ U$, where $\pi$ is the projection map from $\mathbb{C}^2$ to $CP(1)$, that ...
2
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1answer
78 views

Normal ordering of the identity operator

I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator): looking at it from the "Fock space operators POV",the prescription is to move all the ...
2
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2answers
248 views

Do eigenvectors of quantum operators span the whole Hilbert Space?

I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of ...
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2answers
1k views

How To Use Ladder Operators?

I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
2
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2answers
121 views

Dimensionality of Hilbert space

Simple question, but I can't seem to find the answer searching very easily. Does the dimensionality of a Hilbert space correspond to the number of possible states a system can take on? ("The system" ...
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2answers
192 views

Why must quantum logic gates be linear operators?

Why must quantum logic gates be linear operators? I mean, is it just a consequence of quantum mechanics postulates?
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3answers
118 views

How much space to simulate a small Hilbert space?

I'm thinking about trying to do a numerical simulation of some very simple QM problems. How much space do I need? To simulate the Hilbert space? I'd like to eventually simulate the absorption or ...
2
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2answers
454 views

What is the Hilbert space of a single electron?

Is it the same as the space of all possible descriptions of a single electron? If not, how do they differ? Please give the mathematical name or specification of this space or these spaces.
2
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1answer
141 views

Axiomatic structure behind Dirac's formulation of QM?

According to the paper Quantum Mechanics Beyond Hilbert Space by J.P. Antoine, several mathematical structures have been devised to make mathematical sense of Dirac's formulation of quantum mechanics ...
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2answers
135 views

Vector $\vec{z}$ and its conjugate transpose $\overline{\vec{v}^\top}$ - is it the same as $\left|z\right\rangle$ and $\left\langle z \right|$

Lets say we have a complex vector $\vec{z} \!=\!(1\!+\!2i~~2\!+\!3i~~3\!+\!4i)^T$. Its scalar product $\vec{z}^T\!\! \cdot \vec{z}$ with itself will be a complex number, but if we conjugate the ...
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1answer
838 views

Wave function and Dirac bra-ket notation

Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation? How do you get from one to the ...
2
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2answers
333 views

In Dirac notation, what do the subscripts represent? (Solution for particle in a box in mind)

So the set of solutions for the particle in a box is given by $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L}).$$ In Dirac notation $<\psi_i|\psi_j>=\delta_{ij}$ assuming $|\psi_i>$ ...
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2answers
1k views

What's the physical significance of the inner product of two wave functions in quantum region?

I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the inner product of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the ...
2
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2answers
164 views

Harmonic Oscillator - Energy quantisation

The one-dimensional quantum HO can be solved in Schrodinger representation by getting Hermite Differential Equation $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ with solutions $$ y(x) = ...
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1answer
108 views

Resolution of identity in interacting QFT

$\mathbf{Background:}$ Consider a free scalar field $\phi$ ($\mathcal{L}_0 = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{1}{2}m^2 \phi^2$). In the Hamiltonian viewpoint, this system has a ...
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2answers
123 views

Scalar QFT Fock Space

I want to demostrate the following relation of the normal ordered product: $\Omega\equiv:\exp{\left(-\int d^3k~a^{\dagger}(k)a(k)\right)}:=|0\rangle\langle0|.$ I proved the commutation relation ...
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3answers
217 views

Implicit Postulate of Quantum Mechanics

Consider the following quantum system: a particle in a one dimensional box (= infinite potential well). The energy eigenstates wave functions all vanish outside the box. But the position eigenstates ...
2
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1answer
227 views

How can Hilbert spaces be used to study the harmonics of vibrating strings?

The overtones of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1,1/2,1/3,… form the (musical) harmonic series. How can Hilbert spaces be ...
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3answers
554 views

Hilbert space and Lie algebra in quantum mechanics

We are looking for a publication or website that explains the Standard Model in terms of Hilbert space and Lie algebra. We are reading Debnath's Introduction to Hilbert Spaces and Applications and ...
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1answer
98 views

What exactly is a coherent state and why is it interesting?

Please note that I do not have a background in physics, so if possible please refrain from a bunch of $ |x\rangle $ notations, unless clearly specifying what it symbolically means. So I have been ...
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0answers
35 views

Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions ...
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0answers
113 views

What are you studying when you study a Harmonic Oscillator in QM?

This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The ...
2
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1answer
130 views

Non-Hermitian operator with real eigenvalues?

So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian ...
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150 views

Why do some terms vanish in first-order perturbation theory?

In first order perturbation theory, we usually express the first order perturbation in the eigenket of the perturbed Hamiltonian in the basis of the unperturbed Hamiltonian $H_{0}$: ...
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0answers
80 views

What is a continuous superselection sector?

I'm studying the terrible subject of continuous superselection rules and I faced with the following problem. Usually (continuous or discrete) superselection rules are defined involving a direct ...
2
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0answers
100 views

Can we “safely” assume that quantum computing systems will be finite-dimensional?

This is a common assumption in the study of quantum computation to assume that the quantum systems involved are finite-dimensional, since qubits lives in the two-dimensional Hilbert space. According ...
2
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1answer
272 views

Time-ordering in QFT

In Srednicki QFT page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition ...
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3answers
265 views

$\mathrm{Tr}(XY) = \mathrm{Tr}(YX)$?

I'm trying to understand the Dirac notation to understand quantum mechanics better. I'm trying to show the above relation using the Dirac notation. Given $$\mathrm{Tr}(X)~=~\sum_j\langle ...
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4answers
1k views

Necessary condition for square integrable functions?

I'm studying Quantum Mechanics and I came across this which I don't quite understand: For a vector space of functions $f(x)$ to be square integrable (i.e $\int{|f(x)|^2dx < \infty)}$, the necessary ...
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2answers
115 views

In QM, do we deal with basis or orthonormal sets?

Most textbooks say, that given a (countable) basis ${|\phi_n\rangle}$ of a Hilbert space, that every vector $|\psi\rangle$ of the space can be written as: $$\psi\rangle=\sum_{n=1}^\infty ...
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1answer
86 views

Infinite-dimensional Hilbert spaces in physical systems

Can anyone give an example of when infinite-dimensional Hilbert spaces are required to describe a physical system? The standard answer to this question is yes, and I'm sure some of you will be quick ...
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2answers
240 views

General wavefunction and Schrödinger Equation

I'm starting with quantum mechanics and the book I follow (Griffiths) first introduces the wavefunction as the probability density of the position of a 0-spin single particle. Later on I've realized ...
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1answer
106 views

Singular wave function

Given a wavefunction, $\psi(x)$, is it possible for $\psi$ to be singular at a point? Are there any rules against a wavefunctions having any singularities? For instance if $$\psi(x) ...
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1answer
30 views

Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption?

In Dirac's The Principle of Quantum Mechanics he makes the general assumption that "if the measurement of the observable $\xi$ for the system in the state corresponding to $|x\rangle$ is made a large ...
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3answers
88 views

Can one construct a new operator in terms of the powers of another operator?

Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation ...
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2answers
283 views

Hilbert space in quantum mechanics

I think in quantum mechanics we assign to each system a specific Hilbert space i.e. if systems are different then their Hilbert spaces are different. Is this true? If not why? For differernt system I ...
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182 views

The eigenspinors for the spin operator in the $x$-direction?

$$S_x= \frac{\hbar}{2}\quad\begin{pmatrix}0&1\\1&0\end{pmatrix}\quad$$ $$S_x{X_+}^{(x)}=\frac{\hbar}{2}{X_+}^{(x)}$$ How can I find the eigenvalue of $S_x$? My book says $$ \left| ...
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1answer
101 views

Wavefunction's inner product

When two wavefunctions are orthogonal we can write that $$\langle\Psi_n|\Psi_m\rangle=\delta_{mn}$$ This means that $$\langle\Psi_1|\Psi_2\rangle=\langle\Psi_2|\Psi_1\rangle=0$$ But if the two ...
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2answers
312 views

Wave function decomposition

Problem: Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$, find the eigenvalues and eigenfunctions of $\hat{H}$ ...
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1answer
51 views

How to see timelike excitation has a negative norm from the “old covariant quantization”

I have a question in reading Polchinski's string theory vol I p 123, about the "old covariant quantization". It is said ... $\langle 0;k | 0; k' \rangle = ( 2\pi)^D \delta^D (k-k') \tag{4.1.15}$ ...
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1answer
228 views

What does it mean for something to be a ket?

Ok so I will provide the following example, which I am choosing at random from Sabio et al(2010): $$\psi(r,\phi)~=~\left[ \begin{array}{c} A_1r\sin(\theta-\phi)\\ ...
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2answers
332 views

Interference, photon's phase, and the Hilbert space

Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference ...
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2answers
75 views

Unitary transformation between complete + orthonormal bases

Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the transformation matrix $U$: $$ |\psi{'}_n\rangle = U|\psi_n\rangle \\ \langle\psi{'}_n| = ...
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1answer
32 views

Uncertainty Definition QM

On my introductory course in Quantum Mechanics, the uncertainty of an operator $A$ in the state $\psi$ is defined by $$(\Delta A)^2_{\psi}=\langle(A-\langle A \rangle_{\psi})^2\rangle _{\psi}$$ I'm ...
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1answer
152 views

Correspondence between wave function and state vector

I am confused with connection between state $| \psi \rangle$ of a quantum system and corresponding wave function $\psi(x)$ (at a given time). I have been told that for every state $| \psi \rangle$ we ...
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1answer
479 views

Understanding Triplet And Singlet States

We know, $2\otimes 2=3\oplus 1$. Thus we have a spin triplet of states and a spin singlet. Can we regard these states as the spin part of wavefunction for the excited states and the ground state of ...
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3answers
155 views

Normalization of basis vectors with a continuous index?

I have an infinite basis which associates with each point, $x$, on the x-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ ...
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1answer
173 views

Could two different bases of a Hilbert space have different cardinality? [duplicate]

Here is a quote from http://en.m.wikipedia.org/wiki/Hilbert_space#Hilbert_dimension (accessed: Nov. 22, 2013) : As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; ...