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

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2answers
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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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
78 views

Why normalizing spin state is different? Why no need dxdydz?

An electron in a spin state: $$X=A\quad\begin{pmatrix} 2i\\2\end{pmatrix}\quad$$ In order to get A, I have to normalise it, my question is how should write? Since I have to normalise why no need ...
4
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2answers
177 views

An alternative definition of the creation and annihilation operators?

Suppose we have a system of bosons represented by their occupation numbers $$\tag{1} | n_1, n_2, ..., n_\alpha, ... \rangle$$ Then we can define creation and annihilation operators $$\tag{2} ...
4
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2answers
427 views

Quantum Mechanics: Creation and Annihilation operators

Is an eigenvector/eigenstate of the creation operator an eigenvector/eigenstate of the annihilation operator too? Why?
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1answer
111 views

Can general wavefunctions be expressed as kets?

I am confused on bra-ket notation in quantum mechanics. My professor says that a ket is an eigenfunction of some operator. However, for some time now I thought a ket could represent a general ...
4
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1answer
138 views

Hermitian conjugation in Radial Quantization

I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\dagger = t$ and $x^\dagger ...
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1answer
84 views

Lorenz gauge in the Gupta-Bleuler Method

Greiner in his book Field Quantization page 180 & 181 wrote: As shown in (7.24) the Lorenz condition cannot be enforced as an operator identity. Instead we will use it as a condition for the ...
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2answers
243 views

How does the proof of operator commutativity work with non-continuous operators?

In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\{\phi_n\}$, then they commute is given as follows : For any $\phi_n$, $$AB\ \phi_n = a_n\ ...
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0answers
176 views

Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian

Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix ...
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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; ...
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5answers
414 views

Does an electron move from one excitation state to another, or jump?

I'm wondering, when an electron changes state, does it move from one state to another over some (very small) time period? Or does it change from one state to another in no time? If the former, what ...
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2answers
355 views

The harmonic oscillator - ladder operators

Reading from Griffiths. I have got two questions. First, the halmiltonian operator that used to find the energy eigenvalue in only harmonic oscillator is: $$H={\hbar}w(a_-a_+-\frac{1}{2})$$ and ...
16
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2answers
529 views

Hilbert Space of (quantum) Gauge theory

Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down the Hilbert Space of quantum Gauge theory with spin-S. (Are there something ...
2
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1answer
432 views

Why are orthogonal functions and eigenvalues/functions so important in quantum mechanics?

The mathematics and physics we have studied so far at university are heavily focused around the idea of orthogonal functions, orthogonality, sets of solutions, eigenvalues and eigenfunctions. Why ...
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1answer
155 views

Matrix operations on Quantum States in a composite quantum system

Intro (you may skip this if you're an expert, I'm including this for completeness): Say I have two bases for two systems, The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\0 ...
7
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1answer
119 views

Do systems with level crossings have unstable eigenbases?

It's folklore dating back to von Neumann and Wigner that time-dependent Hamiltonian systems tend not to have level crossings of their energy eigenvalues. However, we can of course consider smoothly ...
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2answers
107 views

Angular momentum representation

It is well know that, using position representation $$\langle r\lvert L\rvert \psi\rangle =r \times (-i\hbar\nabla\langle r|\psi\rangle )=r \times (-i\hbar\nabla\psi(r)).$$ However, I read ...
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2answers
109 views

Observable Operator on a Superposition?

I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across. $$ ...
4
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2answers
179 views

The Higgs vacuum

Srednicki's "Quantum Field Theory", an electronic copy of which is freely available here, seems to state on p 205 that the states eq. (32.3) which differ by a phase factor that can range through ...
2
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2answers
927 views

Proving that the hermitian conjugate of the product of two operators is the product of the two hermitian congugate operators in opposite order

I have reach a step in a problem of my quantum mechanics textbook that requires me to prove the following. $$\hat{A}=(\hat{Q}\hat{R})^{\dagger} = \hat{R}^{\dagger}\hat{Q}^{\dagger}$$ I tried to ...
2
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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 ...
2
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3answers
692 views

Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
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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 ...
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6answers
1k views

Why are Only Real Things Measurable?

Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
1
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1answer
451 views

Writing a Wavefunction as a Linear Combination of Eigenstates

We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. ...
2
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1answer
141 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
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1answer
125 views

Shouldn't the addition of angular momentum be commutative?

I have angular momenta $S=\frac{1}{2}$ for spin, and $I=\frac{1}{2}$ for nuclear angular momentum, which I want to add using the Clebsch-Gordan basis, so the conversion looks like: $$ \begin{align} ...
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2answers
244 views

Quantum Mechanics Operator Hermiticity

If $A$ and $B$ are Hermitian operators, show that $$C~:=~i[A,B]$$ is Hermitian too. My work: $$\begin{gather} C=i(AB-BA) \\ \langle\psi\rvert C\lvert\phi\rangle = i\langle\psi\rvert ...
26
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3answers
750 views

A “Hermitian” operator with imaginary eigenvalues

Let $${\bf H}=\hat{x}^3\hat{p}+\hat{p}\hat{x}^3$$ where $\hat{p}=-id/dx$. Clearly ${\bf H}^{\dagger}={\bf H}$, because ${\bf H}={\bf T} + {\bf T}^{\dagger}$, where ${\bf T}=\hat{x}^3\hat{p}$. In this ...
6
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1answer
163 views

Is it always possible to express an operator in terms of creation/annihilation operators?

I'm referring to "Path integral approach to birth-death processes on a lattice", L. Peliti, J. Physique 46, 1469-1483 (1985), available at: http://people.na.infn.it/~peliti/path.pdf The article is ...
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 ...
5
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6answers
1k views

How is quantum superposition different from mixed state?

According to Wikipedia, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state. Now consider state ...
2
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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 ...
1
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0answers
124 views

A basic question about Heisenberg State Kets (in particular the simple harmonic oscillator)

I know base kets in the Heisenberg picture are $U^\dagger |{a}\rangle$ but if the base kets are the base of the hamiltonian, and the hamiltonian is independent of time, are all of the base kets ...
2
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1answer
215 views

Open problem? Square of the wave function $\Psi(x)_{x_o} = \delta(x-x_0)$ of a particle localized at a point $x_0$?

Does anybody know the status of the problem to define the wave function (non-relativistic Quantum Mechanics) of a particle localized at a definite point? Landau-Lifshitz says in chapter 1 that this ...
1
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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 ...
2
votes
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" ...
1
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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 $\hat{H}$ ...
2
votes
1answer
269 views

Continuous spectrum (quantum mechanics) [duplicate]

Does a continuous spectrum of an observable always imply that the corresponding eigenvectors will not be normalizable? If yes, how to prove it?
4
votes
1answer
154 views

Motivation to introduce von Neumann algebras in addition to $C^*$algebras?

Observables are self-adjoint elements of a $C^*$algebra. As such, this structure seems sufficient to describe physics. A theorem by Gelfand and Naimark says that a $C^*$algebra can always be ...
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2answers
330 views

In what sense is the path integral an independent formulation of Quantum Mechanics/Field Theory?

We are all familiar with the version of Quantum Mechanics based on state space, operators, Schrodinger equation etc. This allows us to successfully compute relevant physical quantities such as ...
1
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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 ...
2
votes
1answer
90 views

Is continuous evolution from one eigenstate of operator $O$ to another $O$-eigenstate possible?

Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics. Does this entail that a quantum system cannot continuously evolve from one eigenstate ...
6
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2answers
242 views

Quantum states as rays as opposed to vectors

I recently read that a quantum state is actually defined by a ray and not a vector. That is it is possible to multiply a state $\psi$ by any complex number $c\in \mathbb{C}$ and you won't be changing ...
4
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1answer
133 views

Curved spacetime as a coherent state in string theory

I have a question about Polchinski's string theory book, volume I, p 108. When we write the Polyakov action in curved spacetime, it is said $$ S_{\sigma} = \frac{1}{4\pi\alpha'} \int_M d^2 \sigma ...
7
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3answers
241 views

What are the physical dimensions (units) of the elements in a Hilbert space of a QM system?

In QM, the state vector $|\psi\rangle$ seem to have various dimensions under different representations: (only in space of continuous dimension) $$\langle x|\psi\rangle = [\frac{1}{\sqrt{Length}}]$$ ...
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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 ...
4
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3answers
427 views

How does the momentum operator act on state kets?

I have been going through some problems in Sakurai's Modern QM and at one point have to calculate $\langle \alpha|\hat{p}|\alpha\rangle$ where all we know about the state $|\alpha\rangle$ is that ...
9
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1answer
455 views

The vacuum in quantum field theories: what is it?

In Section 10.1 of his textbook Quantum Field Theory for Mathematicians, Ticciati writes Assuming that the background field or classical source $j(x)$ is zero at space-time infinity, the presence ...