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

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37 views

Are Fock spaces just a special type of tensor algebra?

Are Fock spaces just a special type of tensor algebra? The definitions I am using: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Tensor_algebra From what I can tell, the ...
0
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0answers
31 views

Eigenkets in matrix representation

We use base kets in matrix representation of an operator $X$. Could the base kets be possibly be eigenkets also, of operator X? (Here, I'm taking X as a general operator , not only observable,i.e. ...
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3answers
42 views

Two qubits system in polar co-ordinates

I know that I can write a single qubit state in terms of polar co-ordinates $(r,\theta,\phi)$ on a Bloch sphere. \begin{equation} \rho = \begin{pmatrix} \frac{1+r \cos\theta}{2} &\frac{r \exp(-i\...
4
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1answer
130 views

Coherent states and completeness

Consider one possible definition of a Gaussian (coherent) state in the position representation $$ \langle r | \psi(r_i,p_i) \rangle = \left( \frac{ 2 \gamma}{\pi} \right)^{\frac{1}{4}} \exp \left[ -\...
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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
45 views

What does coherent superposition mean?

There is only one coherent state: $$|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle $$ Also, a pure state does not mean a coherent state. But when does ...
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2answers
40 views

In quantum double slit experiment what is the state vector or density matrix of the electron after the electron passes through the two slits?

Also I would like to confirm my thinking on quantum double slit experiment. Before it passes through the two slits (slit 1 and slit 2), is the electron state vector $\frac{1}{\sqrt{2}}\left(\left|...
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0answers
35 views

Why Does the Dirac delta Function Fix the Normalization of the Basis Vectors in Infinite Dimensions? [duplicate]

On page 60 of Shankar's intro to QM at the very bottom he says that the Dirac delta function fixes the normalization of the basis vectors with an infinite amount of dimensions. I don't understand why ...
27
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5answers
3k views

Why do we need infinite-dimensional Hilbert spaces in physics?

I am looking for a simple way to understand why do we need infinite-dimensional Hilbert spaces in physics, and when exactly do they become neccessary: in classical, quantum, or relativistic quantum ...
4
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1answer
75 views

Is the usage of the Fock space a postulate in QFT?

In this question, when I write Fock space, I mean "the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H", as it is described by Wikipedia. ...
2
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2answers
352 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$\rangle$...
2
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1answer
57 views

How to understand permutations of particles in Quantum Mechanics?

I'm studying identical particles in Quantum Mechanics and I'm having a hard time to understand the idea of permutations of particles from a mathematical standpoint. From one intuitive point of view ...
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1answer
32 views

Normalisation of angular wave function: particle in a circular box

For a particle in a circular box (with radius $R$) with zero potential inside the circle and infinitely high potential outside of the circle, the Schrödinger equation in polar coordinates is: $$-\...
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1answer
170 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 ...
11
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1answer
211 views
+50

Significance of the exception to Gleason's Theorem when n = 2

Gleason's Theorem famously asserts that (appropriately defined) measures on the lattice of a complex Hilbert space can be implemented by density operators via the trace operation, except in the case ...
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1answer
42 views

Want to measure entanglement of the state [on hold]

Good day, I want to measure the state with concurrence and negativity. I do local unitary transformation with represented by $U\in SU(4)$ (Lie group). After the transformation (rotation of angle) ...
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1answer
83 views

Probability in QM: derivation or interpretation? [duplicate]

It is known that coordinates $C_k\in\mathbb{C}$ of the QM-state vectors $|\psi\rangle$ has an interpretation as probability weights $p_k$ in the whole state through the formula like $|C_k|^2=p_k$. We ...
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2answers
106 views

Properties of spectrum of a self-adjoint operator on a separable Hilbert space

So, if I understand it correctly, the spectrum of a self-adjoint operator on a Hilbert space $H$ consists of two parts: $ \newcommand{\ket}[1]{\,\lvert{#1}\rangle} \newcommand{\op}[1]{\hat{#1}} $ ...
6
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3answers
478 views

Hilbert space of a quantum system

The first postulate of Quantum Mechanics as I've learned can be stated as: The states of a quantum system can be described by vectors in a Hilbert space. I've seem also some people also ...
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2answers
74 views

Show that $(x-iy)g(r)$, $z g(r)$ and $(x+iy)g(r)$ are mutually orthogonal [closed]

I want to show that $$\psi_1(x,y,z) = (x-iy)g(r)$$ $$\psi_2(x,y,z) = \sqrt{2}zg(r)$$ $$\psi_3(x,y,z) = -(x+iy)g(r)$$ where $g(r)$ is an arbitrary function of $r = \sqrt{x^2 + y^2 + z^2}$, are ...
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1answer
45 views

Why Does there Have to be Linearity in Ket and Skew Symmetry?

I'm reading Shankar's "Principles of Quantum Mechanics," and on page 8 he states that one axiom in Dirac notation is linearity in ket, and because they are also skew symmetric there is anti-linearity ...
2
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1answer
233 views

What is many-body bound state?

Bound state by definition is a state when particles are bounded together, so then "many-body bound state" would be bound state for a system of many bodies. Then I have several puzzles: is the state ...
2
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0answers
57 views

Rigorous way of box normalisation

This is follow up from an answer to my previous question about unitarity in rigged Hilbert space. As it turns out, that there is no idea of unitarity in rigged Hilbert space (hence no meaningful QM ...
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1answer
59 views

Norm preserving Unitary operators in Rigged Hilbert space

If we take the free particle Hamiltonian, the eigenvectors (or eigenfunctions), say in position representation, are like $e^{ikx}$. Now these eigenfunctions are non-normalisable,so they don't belong ...
13
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2answers
529 views

Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?

When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
2
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2answers
103 views

Does Unitary operator take a pure state to a pure state or can it take a pure state to a mixed state? [closed]

Does Unitary operator take a pure state to a pure state or can it take a pure state to a mixed state? I think so but why? I assume the Unitary operator acts on a pure state only.
3
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2answers
91 views

What kets represent on QFT?

In Quantum Mechanics kets are used to represent states of a system. This is indeed well written in the first postulate of Quantum Mechanics which states that to describe a quantum system we use a ...
1
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0answers
46 views

How can we justify identifying the Dirac delta function with the eigenfunction of position? [duplicate]

I can think of at least two different ways to understand eigenfunctions of operators in quantum mechanics. But neither one seems to provide a good explanation for why we take the position-basis ...
3
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3answers
136 views

Hilbert Space axiom in QM

My question is about the standard axiom on Hilbert's space in orthodoxal QM. It seems that this axiom appeares actually as an external pure mathematical axiom in all textbooks. Say, Mackey introduces ...
0
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2answers
79 views

How can I show that $\mathrm{Tr}\left( f(G^\dagger G)\right)=\mathrm{Tr}\left( f(G G^\dagger)\right)$?

I'm slightly stuck on the following question: Prove that: $\mathrm{Tr}\left( f(G^\dagger G)\right)=\mathrm{Tr}\left( f(G G^\dagger)\right)$ where $G$ is any operator. Using the definition of the ...
11
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3answers
422 views

Can superpositions of baryons with different electric charge and strangeness exist?

I am trying to find out whether the following baryons can exist: $$ |X\rangle = \frac{|u u u\rangle + |d d d\rangle + |s s s\rangle}{\sqrt{3}} $$ $$ |Y\rangle = \frac{|u u u\rangle + |d d d\rangle - ...
1
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1answer
36 views

Dirac notation - trace of product of (bipartite) density matrices

I'm getting confused by the Dirac notation. Suppose I have the following two objects. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...
5
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1answer
172 views

Our choice of basis surely cannot effect possible outcomes of a measurement?

Common sense says that, of course, the outcome of a measurement on a quantum system cannot be affected by what base we choose to represent it in. However, while studying QM text, it seems like they ...
4
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1answer
86 views

Ground state of an adiabatic Hamiltonian as an eigenstate of the total spin

I am going through Quantum Adiabatic Evolution Algorithms with Different Paths by Farhi et al. Here, the authors propose to add a special term to the adiabatic Hamiltonian so that the path of the ...
0
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2answers
57 views

What kind of product is $\prod^n_{j=1}\sigma^{(j)}_x$?

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. The adiabatic Hamiltonian is defined as $$...
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1answer
60 views

Does the Hamiltonian time-evolution operator actually change the state of the system?

According to my understanding of things, the time evolution operator in QM looks something like this, $$U = \exp(-iHt/\hbar)$$ Which acts on the state vector / wave-function of the system to ...
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25 views

Separable and entangled states under unitary transformation

Show that a separable state remains separable under a local unitary transformation. Similarly, an entangled state cannot be turned into a separable state by means of a local unitary transformation....
0
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2answers
35 views

Are negative energy eigenstates orthogonal to plane wave states?

Orthogonality in discrete Hilbert spaces is straightforward - those encountered by typical examples of infinite wells of any type, spin systems etc. Continuous Hilbert spaces are fine too - we ...
3
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5answers
371 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 $| \...
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0answers
23 views

Expand a state in the position basis

Assume we have a state $\rvert\psi\rangle$ in a Hilbert space $\mathcal{H}$. As seen in many introductory texts, the state can be expanded in the position basis as $\rvert\psi\rangle=\int_{-\infty}^{\...
4
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1answer
210 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 $|\...
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1answer
51 views

Product of two Pauli matrices for two spin $1/2$

In the lecture, my professor wrote this on the board $$ \begin{equation} \begin{split} (\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})|++\rangle &= |++\rangle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\...
4
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2answers
56 views

Heisenberg EOM for $\langle x \rangle$ in momentum eigenstate - where is my error?

Equation of motion for expectation value of a quantum particle in a momentum eigenstate: $$\frac{d}{dt} \langle x \rangle = \frac{1}{i h} \langle [x,H] \rangle$$ and since it's in a momentum ...
3
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3answers
4k views

Matrix elements of momentum operator in position representation

I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} \psi(x)=-i\hbar\...
8
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2answers
286 views

Deriving the expectation of $[\hat X,\hat H]$

For a free particle of mass $m$, with Hamiltonian $$\hat{H} = \frac {\hat{P}^2} {2m},$$ where $$\hat{P} = -i \hbar \frac{\partial} {\partial x}.$$ The commutative relation is given by $$[\hat{X}, \...
23
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3answers
4k views

What's wrong with this derivation that $i\hbar = 0$?

Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
0
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0answers
30 views

What is the significance of Dirac ortho-normality? [duplicate]

What is the significance of Dirac ortho-normality? We know for momentum eigenfunction $f(p,x)$ for eigenvalue $p$ , $$\langle f(p',x) | f(p,x)\rangle~=~ \delta(p - p') $$ I am not clear why it is ...
0
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1answer
53 views

Question about Eigenvalues of Hermetian Operators Being Real Numbers

I'm still slogging through Quantum Mechanics: The Theoretical Minimum and I've reached another area that baffles me. Susskind uses the following to show that the eigenvalues of Hermitian operators ...
3
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1answer
131 views

Confusion About Operators

Hello I am currently studying introductory QM and am confused about bases and operators. If I have an operator $\hat{Q}$, does this represent a change of basis matrix? In other words, does $\hat{Q} | \...