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

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Can we have discontinuous wavefunctions in the Infinite Square well?

The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an ...
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2answers
2k 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 ...
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2answers
412 views

Linearity of Quantum Mechanics?

The proof of the No-Cloning Theorem states "By the linearity of quantum mechanics, ..." -- Could someone please give me a rough sketch/outline of what this means? Does it have to do with the Hilbert ...
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2answers
370 views

QM formalism is one big confusion - lack of geometrical explaination with images

I have been trying to learn QM and it went well (all untill harmonic oscilator) until i had to face the formalism: Hilbert space- As a novice to QM i am very sad that in none of the books i have ...
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229 views

Position Representation in Quantum Mechanics

How does the 3d position operator look like in position representation? I know that in 1d the position operator $\hat{x}$ is just multiplication by $x$.
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3answers
635 views

Existence of creation and annihilation operators

In a multiple particle Hilbert space (any space of any multi-particle system), is it sufficient to define creation and annihilation operators by their action (e.g. mapping an n-particle state to an ...
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4answers
270 views

How to apply an algebraic operator expression to a ket found in Dirac's QM book?

I've been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac's book. In the fourth edition, 33rd page, starting from this:$$\xi|\xi'\rangle=\xi'|\xi'\rangle$$ (Where ...
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428 views

Uniqueness of eigenvector representation in a complete set of compatible observables

Sakurai states that if we have a complete, maximal set of compatible observables, say $A,B,C...$ Then, an eigenvector represented by $|a,b,c....>$, where $a,b,c...$ are respective eigenvalues, is ...
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168 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 $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
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224 views

Explanation of Dirac's proof of arbitrary ket being expressible with eigenkets of observable

In P.A.M. Dirac's The Principles of Quantum Mechanics, Chapter 10 (Observables), pp. 40, at the end of the chapter there is a proof that I don't understand at all. Here is a pdf link to the book ...
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697 views

Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...
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187 views

What does $|x⟩|0⟩$ actually mean in bra-ket notation?

Consider the following quote from Wikipedia's page on Shor's algorithm: Initialize the registers to $Q^{-1/2} \sum_{x=0}^{Q-1} \left|x\right\rangle \left|0\right\rangle$ where $x$ runs ...
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2answers
607 views

Observing the exponential growth of Hilbert space?

One of the weirdest things about quantum mechanics (QM) is the exponential growth of the dimensions of Hilbert space with increasing number of particles. This was already discussed by Born and ...
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2answers
109 views

How to associate a Hilbert space with a QM system?

I couldn't really find a fitting title for this question. I'm still relatively new to QM and am trying to get the basics down. I understand that a physical system is associated with a Hilbert Space, ...
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1answer
133 views

What does the notation $\Psi_k/(\Psi_k,\Psi_k)^{1/2} $ mean?

I am currently reading the paper "Gravitation and quantum mechanics for macroscopic objects" by F. Karolyhazy (1966). In his paper, he uses certain notation that I haven't come across before (he also ...
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5answers
351 views

About the definition of expectation value in quantum mechanics

In quantum mechanics, the expectation value of a observable $A$ is defined as $$\int\Psi^*\hat A\Psi$$ But in probability theory the expectation is a property of a random variable, with respect to a ...
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2answers
185 views

Understanding Well Defined States

I am self-studying from a text in QM. Well defined states are mentioned several times. By and large these are consistent and seem to be readily apparent: states of well defined energy are basis ...
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1answer
624 views

How does a state in quantum mechanics evolve?

I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as $$ i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle $$ I am ...
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2answers
229 views

Inner Product Spaces

I am trying to reconcile the definition of Inner Product Spaces that I encountered in Mathematics with the one I recently came across in Physics. In particular, if $(,)$ denotes an inner product in ...
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1answer
3k views

What is an energy eigenstate exactly?

Say you have energy eigenstates \begin{align} \begin{split} |+\rangle= \frac{1}{\sqrt{2}}|1{\rangle}+\frac{1}{\sqrt{2}}|2 \rangle \end{split} \end{align} \begin{align} \begin{split} |-\rangle= ...
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3answers
204 views

Is $0 | \psi \rangle=0$?

For example, the spin operator for spin 1 particle is $\hat{S}_z\doteq\hbar\begin{pmatrix} 1&&\\&0&\\&&-1\end{pmatrix}$ for state ...
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2answers
568 views

Eigenstates of a shifted harmonic oscillator

Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume ...
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500 views

Diagonalization of Hamiltonian

Typically, one way of understanding the physics of an interacting quantum system is by diagonalizing the Hamiltonian. In principle, can we always diagonalize a Hamiltonian, such that it is expressed ...
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275 views

Matrix elements of linear operators - orthonormal basis required?

In an early linear algebra class of mine, I learnt that a linear map $\mathcal{A}$ acting on a vector space could be represented by a matrix $A_{ij}$ according to the rule: $$\mathcal{A}({e_j}) = ...
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1answer
110 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 ...
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1answer
421 views

Example of application of creation/annihilation operators in matrix form

I was wondering how it would sound like the creation/annihilation of particles that we usually do in the context of Dirac formalism, with matrices and vectors. As a reminder we know that: ...
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2answers
139 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. $$ ...
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2answers
1k views

Vector representation of wavefunction in quantum mechanics?

I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
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1answer
200 views

State space of QFT, CCR and quantization, and the spectrum of a field operator?

In the canonical quantization of fields, CCR is postulated as (for scalar boson field ): $$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$ in analogy with the ordinary QM commutation relation: ...
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4answers
140 views

Projection of wavefunction onto basis function

I am given to believe that one way that one would could represent a wavefunction is by the expansion $$\Psi(x) = \Sigma_n \Psi_n(x) = \Sigma_n f_n\phi_n(x) \tag{1}$$ where $\{\phi_n (x) \}$ is an ...
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2answers
207 views

Infinitely many degeneracy of Landau level: Countable or Uncountable?

Description of Landau levels can be found in many standard textbooks of quantum mechanics and here. Two ubiquitous solutions apply either the symmetric gauge $\vec{A}=(-\frac{1}{2}By,\frac{1}{2}Bx,0)$ ...
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429 views

Tensor Product of Hilbert spaces

This question is regarding a definition of Tensor product of Hilbert spaces that I found in Wald's book on QFT in curved space time. Let's first get some notation straight. Let $(V,+,*)$ denote a set ...
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323 views

Banach Space representations of physical systems

I think most physicists mostly model physical systems as some kind of Hilbert space. Hilbert spaces are a strict subset of Banach spaces. Questions: Can physical systems really have non-compact ...
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1answer
703 views

Quantum mechanic newbie: why complex amplitudes, why Hilbert space? [duplicate]

I'm just starting learning quantum mechanics by myself (2 "lectures" so far) and I was wondering why we need to define quantum states in a complex vector space rater than a real one? Also I was ...
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1answer
55 views

Eigenvectors of $p_x$ in a particular domain

Defining the $p_x$ operator for the problem of particle in a infinite well. In the book by Capri on Quantum mechanics, the domain of the operator is given by, $$ p = -i\hbar \frac{\partial ...
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1answer
112 views

States in light cone string theory

Currently I'm trying to understand string theory in the light cone quantization. I just have had a look into Polchinski (Vol. 1, Introduction to the bosonic string), because – as far as I could see – ...
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2answers
79 views

Query on an operator acting on a function

I have a naive question about an operator acting on a well-behaved function. Let us say, we are talking about space translation operator acting on a function $\psi(x)$: ...
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2answers
163 views

What is the precise definition of state of a quantum system?

On studying quantum states of the hydrogen atom, I came with a question: what is the precise definition of a quantum state? Particularly, when solving the Schroedinger eq for the H using separation of ...
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1answer
85 views

Eigenvalues of Infinite Dimensional Matrix [duplicate]

If I take a infinite-dimensional square matrix, what can I say about its eigenvalue spectrum? Will they have a discrete infinity of eigenvalues or continuous infinity of them?
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2answers
187 views

Space translation of operators, states, and particle densities

In Sidney Coleman's Lectures he talked about space translations such that $$\tag{1} e^{ia P}\rho(x) e^{-ia P} ~=~ \rho(x-a),$$ but when I expanded the exponentials and used the commutation relation ...
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2answers
139 views

Are locality and separability two distinct notions?

Is there any difference between locality and separability in quantum mechanics, or do they mean the same thing? It seems authors do not always agree.
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207 views

Quantum Mechanics Basics: product space

Consider a coupled harmonic oscillator with their position given by $x_1$ and $x_2$. Say the normal coordinates $x_{\pm}={1\over\sqrt{2}} (x_1\pm x_2)$, in which the harmonic oscillators decouple, ...
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2answers
89 views

Why is the “complete metric space” property of Hilbert spaces needed in quantum mechanics?

I have been learning more about Hilbert spaces in an effort to better understand quantum mechanics. Most of the properties of Hilbert spaces seem useful (e.g. vector space, inner product, complex ...
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1answer
84 views

Self-adjoint and nonpositive differential operators

I recently tumbled over a statement in a geophysics paper (PDF here). They have a wave equation which they formulate as $$ \frac{1}{v_0}\frac{\partial^2}{\partial t^2} \begin{pmatrix}p \\ ...
3
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1answer
756 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 ...
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1answer
260 views

Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $

I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that $$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
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195 views

The state of Indefinite metric in Quantum Electrodynamics

I faced difficulties to grasp why indefinite metric is introduced from no where in QED, after searching internet I found that this is a problem in QED, because one needs it to preserve theory's ...
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96 views

Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
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46 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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195 views

Is $∣1 \rangle$ an abuse of notation?

In introductory quantum mechanics it is always said that $∣ \rangle$ is nothing but a notation. For example, we can denote the state $\vec \psi$ as $∣\psi \rangle$. In other words, the little arrow ...