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

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9
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4answers
512 views

Curvature of Hilbert space

That may appear as a dumb question, but: Does Hilbert space have curvature, or is it a flat space? How and why?
-2
votes
0answers
46 views

Expectation value of an Observable and Eigenstates

I am learning about Quantum Mechanics at the moment and I was wondering about Eigenfunctions and Observables. The question I would like to ask is, If a wavefunction is not an eigenstate of an ...
1
vote
0answers
40 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
votes
0answers
33 views

Eigenkets in matrix representation [on hold]

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. ...
2
votes
3answers
57 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\...
1
vote
1answer
52 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 ...
0
votes
2answers
43 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|...
0
votes
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 ...
4
votes
1answer
77 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
votes
1answer
61 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 ...
4
votes
1answer
136 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[ -\...
0
votes
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: $$-\...
0
votes
1answer
44 views

Want to measure entanglement of the state [closed]

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) ...
1
vote
1answer
85 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 ...
0
votes
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 ...
7
votes
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}} $ ...
0
votes
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 ...
11
votes
1answer
234 views

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 ...
2
votes
0answers
58 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 ...
2
votes
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 ...
3
votes
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
vote
0answers
48 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 ...
0
votes
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 ...
3
votes
3answers
139 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 ...
1
vote
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. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...
0
votes
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 ...
0
votes
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 $$...
-1
votes
0answers
26 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
votes
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 ...
0
votes
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}^{\...
1
vote
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
4
votes
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 ...
3
votes
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} | \...
0
votes
1answer
31 views

Norm of quantum state in three dimensions

The Born interpretation states that for a particle with a wave function $\Psi(x)$, the total probability of finding that particle at some point in space is equal to $\int_{-\infty}^{\infty}\Psi(x)^*\...
3
votes
2answers
81 views

Understanding operator bra-ket notation

Hi I have a question that might be a bit trivial. I have just completed learning a section on the bra-ket notation. There is a statement that the following is prohibited $$\hat{A}\langle\psi|, ~|\psi\...
0
votes
1answer
46 views

Verification of proof of complete set of commuting operators

Hi I am interested in the validity of the following proof. I am interested in the validity of this particular proof as I am aware of how to prove this result in a different way. Theorem: If two ...
2
votes
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.
0
votes
1answer
64 views

Definition of the S-matrix

when I think about scattering process I reach to slightly another definition to the S-matrix. because I understand my reasoning I hope someone could refine it to a correct one so that I can have a ...
0
votes
1answer
65 views

Bloch sphere representation of $\sigma_x$ operator on $|1\rangle$

I am trying to visualize a Hamiltonian H=$\hat{\sigma_x}$ $$ \hat{\sigma}_{x} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) $$ acting on the state $| 1 \rangle$. I can write ...
3
votes
4answers
324 views

Can every density operator be written as an outer product of two vectors?

I have a feeling this is a very basic question. I apologize if it is. Using Dirac's notation, can every (mixed) density operator $\rho_A$ of system $A$ be written as the ket-bra (outer) product $|a_1 ...
5
votes
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 ...
0
votes
0answers
29 views

Homework question on the Poynting vector

I need to show what the Poynting vector becomes when i insert the definition of A through creation and annihilation operators, what i get is the expected answer plus a term containing two creation ...
0
votes
1answer
42 views

Global and relative phases of kets in QM

In one of the questions I'm trying to solve it is asked to, first, compute probabilities for the respective results of the Stern-Gerlach measurements performed on each state $\lvert\psi_1\rangle$, $\...
4
votes
1answer
98 views

Dirac Equation in RQM (as opposed to QFT) is written in which representation?

In introductory Quantum Mechanics treatments it is common to see the Schrödinger's equation being written, simply as: $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},t)+V(\mathbf{r})\Psi(\mathbf{r},t)=...
3
votes
5answers
194 views

Where does $\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$ come from?

It's a very basic question, where does the relation $$\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$$ for any square integrable $\psi(x)$ come into existence? Some texts I found states that the above ...
3
votes
1answer
64 views

Vacuum expectation value in presence of a source

If a vacuum is translationally invariant i.e., $P^\mu|0\rangle=0$ or $e^{(\pm ip\cdot x)}|0\rangle=0$, we can express the the vacuum expectation value of a field as $\langle 0|\phi(x)|0\rangle$ as $$\...
-6
votes
1answer
62 views

Is superposition state of SHO ever observed? [closed]

Feynman says, "It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong." So, is superposition state of Simple ...