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

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1answer
49 views

Second Quantization: The Identity Operator does not Commute?

Let me take the simplest possible example. Consider the fermonic Fock-space $\Lambda^*(\mathbb{C}^n)$ built out of a finite-dimensional, oriented single-particle Hilbert space $\mathbb{C}^n$ with ...
2
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1answer
122 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
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3answers
2k views

Quantum Joke (not a real joke, not a riddle)

Supposing I want to make a quantum joke, like writing this on a coffee machine: $$| \text{Status}\rangle = \frac{1}{\sqrt{2}}\ \big( | \text{Working}\rangle \color{red}{\pm} | \text{Down}\rangle \...
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2answers
111 views

Normalisation of free particle wavefunction

The wavefunction $\Psi(x,t)$ for a free particle is given by $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$ This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
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1answer
130 views

Expectation value of an operator and commutator relation

I have a quantum operator $A.$ It's expectation value is constant respect to time. I mean $$\langle \psi(t)|A|\psi(t)\rangle$$ is a constant values. If I know $|\psi(t)\rangle$ is not an eigenstate ...
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1answer
103 views

Inserting the resolution of identity correctly

In a text on path integrals, I find the following: \begin{equation} \langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|q_j\rangle = \int\frac{dp}{2\pi}\langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|p\rangle\...
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1answer
88 views

Momentum in quantum harmonic oscillator with step up and step down operators [closed]

I'm hitting a wall in my understanding of the momentum operator in a quantum harmonic oscillator. I've showed that $p = (a^\dagger - a)\sqrt{\frac{m w \hbar}{2}}i$ where $a^\dagger$ and $a$ are the ...
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2answers
202 views

What is a Hilbert Space?

The Hilbert Space is the space where wavefunction live. But how would I describe it in words? Would it be something like: The infinite dimensional vector space consisting of all functions of ...
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2answers
71 views

Simple question: Euclidean versus Hermitian form

This may be a basic question, but why is the inner product of bra and ket Euclidean inner product [link] and not more general Hermitian form? [link] Is there something fundamental stating that $M$ ...
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1answer
43 views

Parity operator eigenstates [closed]

I have a problem I cannot solve on my own. I have given two states $\psi_1$ and $\psi_2$ and an Operator $O$ such that $P \psi_1 = \epsilon_1 \psi_2$, $P \psi_2 = \epsilon_2 \psi_2$ and $POP^{-1} = ...
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1answer
38 views

State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned State operator corrponding $i.e$ $S\times S$ to $R^2$ which was wrong question. By studing Ginsparg's ...
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1answer
153 views

In nonrelativistic Quantum Mechanics, is the expectation value of a sum of operators always equal to the sum of the expectation values?

Suppose that $\lvert \psi_n \rangle$ are the eigenvectors of a Hamiltonian, $\hat{H}$, which span some Hilbert space $\mathcal{H}$ and satisfy $$\hat{H}\lvert \psi_n \rangle = E_n \lvert \psi_n \...
3
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0answers
80 views

Picture-independence of quantum mechanics

I've been thinking about the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics in the following terms lately: a quantum system is a Hilbert space $\mathcal{H}$ equipped with ...
0
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2answers
134 views

Coefficients and wavefunction in quantum mechanics

In general quantum mechanics we represent the state of a system with a state vector $| \psi \rangle $ in some Hilbert space in some base. Assuming a complete discrete set of bases vectors $ |n \rangle ...
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2answers
47 views

Probability that a measurement will be in some set

Let $\mathcal{H}$ be the Hilbert space of a quantum system and $A$ one observable in $\mathcal{H}$. If $A$ has discrete spectrum $\{a_n : n \in \mathbb{N}\}$ for simplicity, then by the postulates of ...
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5answers
190 views

Looking for clarification on superposition [closed]

I have always had a hard time accepting the concept of superposition from quantum mechanics. I know that the leading physicists say that the cat is both alive and dead until it is observed and that an ...
4
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3answers
306 views

Dirac notation - specific acting orientation for operators

I have this doubt: Imagine two operators $A$ and $B$ and the state $\psi$. I know that the following statement is true: $$\langle\psi| A|\psi\rangle^*=\langle\psi| A^\dagger|\psi\rangle$$ But ...
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0answers
12 views

Numeric fermiomic creation operators and unit cell

I have to do some numerics (e.g. FFT in Maple/Octave) on a 1D fermionic chain without forces between the particles. The description says that two sites build an unit cell. What does this mean? And ...
2
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1answer
313 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 ...
2
votes
2answers
70 views

Expectation value of an imaginary operator acting on a real function

In a video (http://youtu.be/r_gBQ_qhg8U?t=9m58s) it's stated that a matrix element of an imaginary operator acting on a real wave function is zero, i.e. $$\langle\text{real}|\text{imaginary}|\text{...
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1answer
135 views

What does a line above a commutator, e.g. $\overline{[x, H]}$ mean?

What does this notation mean in relation to quantum mechanics? $$\overline{[x,H]}\qquad\text{or}\qquad\overline{[p,H]}\tag{1}$$ I know $[x,H]$ is just the commutator e.g $xH-Hx$, and the anti-...
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2answers
65 views

Show that $|n\rangle$ is correctly normalized [closed]

Prove that $$|n\rangle = \frac1{\sqrt{n!}} (\hat a^\dagger)^n |0\rangle$$ is correctly normalized. I know I must show its bra-ket equals 1 but I don't know what bra-ket notation really means, so I ...
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0answers
132 views

Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
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1answer
102 views

Closure relation for degenerate eigenkets

Consider an observable in quantum mechanics, with a degenerate eigenvalue in a continuous spectrum. Is it possible for such an eigenvalue to have a finite degeneracy? If the degeneracy is infinite, ...
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2answers
102 views

Griffiths use of a linear transformation on basis vectors. Need help with derivation

In Griffiths' intro to Quantum Mechanics 2nd edition, in the appendix A.3 which is a review on linear algebra and matrixes (on page 441), he states that a linear transformation on a set of basis ...
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1answer
161 views

Is there a mathematical basis for Born rule?

Wave function determines complex amplitudes to possible measurement outcomes. The Born Rule states that the probability of obtaining some measurement outcome is equal to the square of the ...
3
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0answers
80 views

Underlying C*Algebra operators in standard quantum mechanics?

Linearity in standard quantum mechanics (QM) is the key to making the math possible in this field, but the presence of nonlinear operators in QM is what is more generally dealt with. Working with the ...
3
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1answer
146 views

Why, in quantum field theory, is $\hat{a}(p)|0\rangle=0$?

My Quantum Field Theory lecturer just said that $\hat{a}(p)|0\rangle=0$ because the vacuum state contains no particles. Now, according to Wikipedia, "according to quantum mechanics, the vacuum ...
5
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3answers
196 views

Inverse Quantum Operator

In the quantum harmonic oscillator problem, how would one go about calculating $$\left\langle n\left|\frac{1}{X^2}\right|n\right\rangle$$ using raising and lowering operators $a^{\dagger}, a$ only, ...
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2answers
219 views

Aren't all spin states (up, down, left, right, in, out) orthogonal?

Hopefully someone can clear up a basic misconception I am having about the nature of spin state vectors. According to the book i am reading, The basis vectors of up/down spin are orthogonal to each ...
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1answer
52 views

Show that translation and rotation operator are unitary

I have a problem understanding how to show that operators are unitary if they are not in the "normal" matrix form. The translation operator is defined as $$(T_v \psi)(x) = \psi(x-v)$$ and the ...
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1answer
82 views

Which definition of a quantum field is right?

In introductory quantum field theory, I was taught that, given a single-particle Hilbert space $\mathcal H$, the quantum field operator for that type of particle was a mapping $\varphi(x)$ from $k$-...
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1answer
69 views

Transformations of states in quantum mechanics

In Classical Mechanics we usually describe the possible configurations of a system by points on a smooth manifold $M$ which is the configuration manifold of the system. In that case, when we talk ...
2
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2answers
125 views

Does every Hilbert Space carry a representation of Poincare group?

We know all infinite dimensional Hilbert Spaces are unitarily equivalent. It should follow therefore that if I have an unitary representation of say Lorentz or Poincare group on one infinite ...
2
votes
1answer
114 views

Why are eigenspaces of a Hermitian operator mutually orthogonal? [closed]

In Quantum Mechanics, from the properties of the solution of Schrodinger's Equation inside the infinite well, is that they are: Mutually orthogonal for different eigenvalues. Orthonormal. Complete. ...
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2answers
153 views

General formula for expanding wave function in terms of orthogonal states?

Given a wave function $\psi(x) = \langle \psi | x \rangle$. It can be expanded in terms of orthogonal states: $$ \langle \psi | x \rangle = \sum_n \langle \psi | n \rangle \langle n |x \rangle $$ ...
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1answer
57 views

Unitary Transfomation from One Basis to Another [closed]

So we have two orthonormal linearly independent basis $\{ |\phi_1 \rangle, \dots, |\phi_n \rangle \}$ and $\{ |\psi_1 \rangle, \dots, |\psi_n \rangle \}$. We can express the basis vectors of the ...
0
votes
1answer
70 views

Quantum States, Hilbert Space and Time

I'm having troubles with the assertion "(normalizable) wave-functions constitutes (projective) Hilbert space". The standard argument I find for this seems to go as following: say $\Psi(\vec{x},t)$ is ...
0
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2answers
39 views

Proportionality of states in quantum harmonic oscillator

What is the justification for $a_{\pm} \psi_{n}$ being proportional to $\psi_{n\pm1}$ in a quantum harmonic oscillator? Here $a_{\pm}$ is the raising/lowering ladder operator.
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0answers
46 views

Why can differentiating a function carry it out from Hilbert space?

I was just doing a QM Griffiths Problem. I was able to get it correct, but I have a few questions. Let $f(x)=x^v$ be defined on $[0,1]$ where $v= 1/2$ then $df/dx = \frac{x^{-1}}2$ Then, we know ...
2
votes
1answer
172 views

Proving polarization identity for operators in complex vector spaces

In general for two operators to be equal, all their (matrix) elements must be equal $$A = B \rightarrow \langle \phi_1|A| \phi_2\rangle=\langle \phi_1|B| \phi_2\rangle$$ However, I am asked to show ...
0
votes
2answers
132 views

$(H\Psi(x,t))^*=H\Psi^*(x,t)$?

In the solutions of an exercise I got confused about the following equality $$(H\Psi(x,t))^*=H\Psi^*(x,t).$$ Is this true in general? Or in special cases? It seems to imply that H is a real matrix ...
6
votes
2answers
256 views

What is the connection between Hilbert Space and path integrals?

Given a space of states $|\rangle$, $|x\rangle$, $|x,y\rangle$, with the creation operators such as $\hat{\phi}(x)|y,z\rangle=|x,y,z\rangle$ for creating a particle at position $x$ and so on. How ...
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2answers
132 views

Why/How do the coefficients associated with atomic orbitals superposed to form hybrid orbital determine their spatial orientation?

In my previous Phys.SE question, I asked for why $ \newcommand{\k}[2]{\langle #1|#2 \rangle} c_1,c_2,c_3,c_4$ in $$\psi_{sp^3}= c_1\psi_{2s}+ c_2\psi_{2p_{x}} + c_3\psi_{2p_y}+ c_4\psi_{2p_{z}}$$ ...
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1answer
110 views

Matrix represenation of total angular momentum operator

I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively. For the total angular ...
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2answers
130 views

Does this quote from my textbook imply that not all states are superpositions?

I read this at a book; The difference between bits and qubits is that a qubit can be in a state other than $|0\rangle$ or $|1\rangle$. It is also possible to form linear combinations of states,...
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2answers
345 views

Problem in understanding the concept of 'superposition' as explained by Dirac

The concept of 'superposition' has really made me insane, actually. What I thought it was just simple superposition of matter waves. For instance, let's take the Double-Slit experiment: take the ...
0
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2answers
113 views

Relationship between nodes in wavefunction and orthogonality

I read that if I want to construct a wavefunction orthogonal to given $n$ orthogonal wavefunctions, then the new wavefunction should have $n$ nodes. Is this valid under all conditions? Is there a ...
0
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3answers
441 views

Is the superposition of stationary states a stationary state? If not, then why not?

I am a beginner in Quantum mechanics and as I understand,the superposition of stationary states is also a solution of time-independent Schrödinger equation (TISE). The wave functions that are the ...
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0answers
107 views

Problem in understanding Feynman's explanation of the Dirac-Delta function

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...