All Questions
Tagged with definition hilbert-space
77
questions
0
votes
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107
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Mathematical meaning of a position eigenbra $\langle x_0 |$
Let $|x_0\rangle$ be an position eigenket. The physical picture I have for $|x_0\rangle$ is a particle located at $x_0$. Thus it should be represented by a delta function $\delta(x-x_0)$.
For $f\in L^...
- 3,350
1
vote
1
answer
75
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Difference between stationary states, collision states, scattering states, and bound states
A few weeks ago, I was presented one-dimensional systems in my QM class, and of course one-dimensional potentials too. Nonetheless, I'm still a bit unclear about the terminology my professor uses. ...
- 1,616
-1
votes
2
answers
99
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Are Hermitian operators Hermitian in any basis? [closed]
Given a Hilbert space and a Hermitian operator defined on it, will the operator exhibit Hermiticity in any basis used to span the space? My thought on this is that this must be the case, after all, if ...
- 4,399
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votes
1
answer
79
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What is the difference between $(\mathcal{H}\setminus \{ 0\})/\mathbb{C}^*$ and $\mathcal{H}_1/U(1)$?
Let $\mathcal{H}$ be a Hilbert space. We define the projective Hilbert space $\mathbb{P}\mathcal{H}$ as $\mathcal{H}\setminus \{ 0\}/\mathbb{C}^*$. Then $[\Psi]=\{ z\Psi :z\in \mathbb{C}^*\}$.
On the ...
- 374
2
votes
0
answers
115
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What is the definition of bound state in quantum field theory?
I asked a question a while a go what is a bound state and the question was closed because there is a similar question.
Now since best description we have to describe nature in quantum field theory
How ...
- 445
-2
votes
1
answer
144
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What is meant by " a basis is diagonal"?
I am trying to understand Schmidt decomposition. I am stuck in one sentence here. See the example picture.
Here, I can understand everything except the line "For both
HA and HB the Schmidt basis ...
1
vote
4
answers
512
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What actually is superposition?
What does superposition actually mean? Can something like an atom actually be in two different states at once or do we just not know which state it is in? Also, how can our act of observing something ...
- 55
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2
answers
1k
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Tensor Product vs Direct Product in QM
Consider adding angular momentum. Shankar describes the state of the system as the direct product of states while Ballentine (and I think most other people) describes the state of the system as the ...
- 69
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vote
1
answer
120
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What is the definition of a stationary state?
In this answer, a state, $\psi(t)$ is said to be stationary if
$$
\begin{equation*}
|\psi(t)|^2=|\psi(0)|^2.
\end{equation*}
$$
That answer then concludes that a state can only be stationary if it is ...
- 239
1
vote
1
answer
132
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A theorem about functions of self-adjoint operators
It is very common (see e.g. page 18 of Ballentine's Quantum Mechanics: A Modern Development) for the following development to take place. We couch the discussion in Dirac's bra-ket notation noting ...
- 1,095
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2
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378
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How does one write Adjoint, Self-adjoint and Hermitian operators in Dirac notation?
The following portion is paraphrased from Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence.
The adjoint of a linear operator $\hat{A}$, denoted by $A^\dagger$, is an ...
- 11.8k
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votes
1
answer
126
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How can eigenstates of a hermitian operator be orthogonal without explicitly defining the inner product?
It's a well known fact that for any hermitian operator, say $H$ (assuming there is no degeneracy), $${\left< a_i \right.\left| a_j \right> \over \sqrt{\left< a_i \right.\left| a_i \right>...
- 317
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votes
0
answers
74
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What is a state space in quantum mechanics?
I have begun reading chapter 11 of Zwiebach's "A First Course in String Theory" 2nd edition. Section 11.2 deals with the Heisenberg and Schrodinger pictures. Both pictures will use the same &...
- 246
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votes
1
answer
330
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The name of the Hilbert space in quantum mechanics
I know that states in quantum mechanics are positive trace class operators acting on a separable complex Hilbert space $\mathcal H$ and having trace = 1. Specifically, pure states are one-dimensional ...
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votes
1
answer
290
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Mathematical definition of annihilation and creation operators
I am self-studying quantum field theory and have gotten to creation and annihilation operators, respectively denoted $A^\dagger$ and $A$. Conceptually I understand what these objects are, at least on ...
- 3,350
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30
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Differences between energy level and end energy state [duplicate]
What is differences between energy level and end energy state in quantum mechanic?
Are they same?
Is the energy state same as the quantum state?
0
votes
2
answers
131
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Question on Dirac notation with operator [closed]
What does $\langle\psi|A|\phi\rangle$ mean if $A$ is some operator like how does $A$ acts on these two vectors $\phi$ and $\psi$ and what is it equal to and also does $A$ act on both vectors or just ...
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2
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627
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What is the difference between "cluster states" and "graph states"?
I wonder about the difference between the cluster state and the graph state.
I guess the only difference is the graph of the cluster state is limited to a two-dimensional square lattice
The concept of ...
- 51
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votes
1
answer
364
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Radial position operator
While trying to find the expectation value of the radial distance $r$ of an electron in hydrogen atom in ground state the expression is:
$$\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \...
- 1,270
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vote
1
answer
513
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Definition of a wave packet
In Shankar's QM book page 168, the author stated
a wave packet is any wave function with reasonably well-defined
position and momentum.
What does he mean by resonably well-defined position and ...
- 4,235
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votes
0
answers
54
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Inner product evaluation in QM
On wikipedia on the page for inner product it states that for any two $x,y$ in a vector space $V$ the inner product $(\cdot , \cdot)$ satisfies $(ax, y) = a(x,y)$ where $a\in\mathbb{C}$.
The inner ...
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1
answer
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What does Leggett mean by quantum states like $|\psi\rangle=(a|\psi_1\rangle+b|\psi_2\rangle)^N$?
In his article (p. 1986) Legett uses the notation $|\psi\rangle=(a|\psi_1\rangle+b|\psi_2\rangle)^N$ to classify "macroscopic quantum phenomena". Does the "$^N$" mean "$\...
- 448
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3
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529
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$\left\langle{\hat{O}}^\dagger\varphi\middle|\psi\right\rangle$ How do I act the operator in bra?
$$\left\langle\varphi\middle|\hat{O}\middle|\psi\right\rangle=\left\langle{\hat{O}}^\dagger\varphi\middle|\psi\right\rangle.$$
In above formula, I have confused what does mean $\left\langle{\hat{O}}^\...
- 75
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1
answer
441
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What is the difference between an eigenfunction and a wavefunction?
This question is an additional point of clarification to my previous question about adding position and momentum eigenstates.
For simplicity, suppose I had a particle in an eigenstate of momentum, $|p\...
- 2,680
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vote
1
answer
106
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Action of permutation operator on other operators
I'm watching MIT 8.06 Quantum physics, lecture $23.2$ See for example [1] Particularly See $5:41$. It is shown that
$$P_{21}B(1)P^\dagger_{21}|u_i\rangle_1\otimes |u_j\rangle_2=|u_i\rangle_1\otimes |...
- 11.9k
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votes
1
answer
443
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What is a quasibound state and how is it different from a bound state?
What is a quasibound state and how is it different from a bound state?
I have read this term in nuclear physics in the context of compound nucleus formation. A compound nucleus $C$ is formed by the ...
- 11.8k
1
vote
1
answer
182
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Requirement of Jordan-Wigner string in creation operator on Fock state
Our lecture notes described the action of the particle creation operator on a fermionic Fock state:
$$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$
I am ...
- 2,604
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votes
1
answer
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What exactly is a Fock state?
I am a bit confused by the way a Fock state is defined and hope to find some clarification.
The Fock space is defined as the direct sum of all $n$-particle Hilbertspaces $H_i$
$$F = H_0 \oplus H_1 \...
- 387
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votes
1
answer
116
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Is this definition of the Fourier Transform of a quantum field operator rigorous?
Let there be a a quantum field operator $\hat\phi(t,\vec{x})$ which, because it acts (pointwise) on a separable Hilbert space, I expect I can write as
$$\hat\phi(t,\vec{x}) = \sum_n\sum_m\phi^n_m(t,\...
- 34
3
votes
3
answers
430
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Understanding the bra-ket antilinear correspondence
I can't follow how the above argument leads to (1.8).
I am able to prove it only if I can show $$\langle a | c\rangle+\langle b| c\rangle=(\langle a|+\langle b|)\,|c\rangle$$
But I don't understand ...
- 1,270
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votes
1
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437
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Bra-Ket and inner products
We denote a scalar product of two vectors $a, b$ in Hilbert space $H$ as $(a,b)$ or $\langle a, b\rangle$.
In Bra-Ket notation, we denote a vector $a$ in Hilbert space as $|a\rangle$. Also, we say ...
- 1,270
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1
answer
260
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What does "operators on a Hilbert space form an algebra" mean?
I was reading some group theory notes and I am familiar with the concept of a Lie algebra, but I cannot imagine what the following formulation means:
What is more, not only states, but also the ...
- 183
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1
answer
58
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Doubt about property of hermitian operator
For any hermitian operator M, prove that
\begin{equation}
\langle Ma|b \rangle = \langle a|Mb \rangle
\end{equation}
My attempt:
Let
\begin{eqnarray}
\langle a| = \sum_i a_i^*\langle i|\\
|b\rangle = \...
6
votes
1
answer
545
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Definition of the $S$-Matrix in Schwartz QFT-Book: Why is $\langle f, t_f | i, t_i \rangle$ in the Schroedinger picture, and not Heisenberg-picture?
On page 51, (equation 5.1), Mathew Schwartz introduces the $S$-matrix as
\begin{align}
\langle f| S | i \rangle_{Heisenberg} = \langle f, \infty | i, -\infty \rangle_{Schrödinger}
\end{align}
Were $|i,...
- 6,763
2
votes
2
answers
900
views
Domain of an adjoint operator
I'm studying a bit of functional analysis for quantum mechanics and I'm stuck on a definition our professor gave us.
Given an operator and its domain $(A,\mathcal{D}(A))$ densely defined in $\mathcal ...
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votes
3
answers
337
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What is the meaning of the ket states in the notation $\langle x_f,t_f|x_i,t_i\rangle$?
Path-integral amplitudes are denoted by the inner product $\langle x_f,t_f|x_i,t_i\rangle$ where $|x_i,t_i\rangle$ is a time-independent position eigenstate of the time-dependent Heisenberg picture ...
- 11.8k
7
votes
1
answer
681
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Why do self-adjoint operators have to be densely defined?
I have been watching the Schiller lectures on QM and have been going through ‘quantum mechanics and quantum field theory’ by Dimock.
Both seem to ensure operators are densely defined, especially if ...
- 1,933
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votes
1
answer
189
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Bra-representation in quantum mechanics
I'm a bit confused with the 'bra' notation in the representation of the Schrodinger equation. For example, in the momentum representation, the state $|E_{n}\rangle$ is represented by the function $\...
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votes
1
answer
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What is a neutrino state if not a particle?
When reading about the 2015 Nobel prize and how this led to the possibility of the existence of sterile neutrinos I am told that:
"(...) three active neutrinos $\nu_e$, $\nu_\mu$, $\nu_\tau$, are ...
- 753
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votes
1
answer
772
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What does it mean for a wave function to be "bounded" while imposing regularity conditions?
This question is more like a definition-confusion which is causing me to misunderstand several things. So, I am taking the MIT 8.05 Quantum Physics-II course and the instructor while mentioning the ...
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votes
1
answer
199
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Correspondence between mathematician's and physicist's vertex operator algebra (VOA)
I have some conceptual doubts to clear up, in terms of piecing together what we learn of a vertex operator algebra (VOA) in conformal field theory, and how it is defined by a mathematician, say from ...
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3
answers
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Confused about definition of three dimensional position operator in QM
My QM text defines the position operator as follows:
The position operator $X= (X_1,X_2,X_3)$ is such that for $j=1,2,3: \ X_j \psi(x,y,z)= x_j \psi(x,y,z)$.
To me this can mean two things.
1) $...
- 151
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votes
2
answers
677
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Are all bound states normalizeable?
Following Griffiths eq. (2.91) on p. 52 one may define a bound state to be an energy eigenstate
$$H|E\rangle=E|E\rangle\tag{1}$$
with an energy being smaller than the potential far away from the ...
user224659
2
votes
2
answers
1k
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Physical meaning of Transpose of an Operator in Quantum Mechanics?
What's the physical meaning of transpose of a matrix in Quantum Mechanics?
Although for Unitary or Orthogonal operators, I know that transpose of that operator would reverse the action and that's ...
- 175
4
votes
2
answers
500
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Completeness of Norm in Hilbert Space
I am not sure what it really means for the norm to be complete in a Hilbert Space. Can you provide me a proper definition? I am aware of the formula $||\Psi|| = <\Psi|\Psi>^{1/2}$.
What are ...
- 43
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vote
2
answers
576
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Clarification in the difference between metastable states and excited states
The answer of this question What is the difference between metastable states and excited states?
is that the difference lie in the the time that the systems lie in a given state.
So for example take ...
- 445
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votes
1
answer
716
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What is the difference between metastable states and excited states?
In the book Mathematical concepts of quantum mechanics ,Stephen J. Gustafson Israel,Michael Sigal, they say
The notion of a resonance is a key notion in quantum physics. It
refers to a metastable ...
- 445
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votes
1
answer
5k
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Difference between Fock space and Hilbert Space
I am beginner in QFT. I would like to know why there is a need of constructing Fock space for a $N$-particle system? Why can't we represent this many body system just as the tensor product of Hilbert ...
- 1,348
5
votes
2
answers
4k
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What does it mean when a degeneracy is lifted?
I would like to ask what is the meaning of degeneracy been lifted? For example when the Schrodinger equation is subjected to magnetic field, there is a $m\ell$ degeneracy is lifted while $\ell$ ...
8
votes
3
answers
385
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Equivalent definitions of total angular momentum
Consider the equality
\begin{equation}\exp\left(-\frac{i}{\hbar}\boldsymbol{\phi J}\right)\left|x\right>=\left|R(\phi)x\right>,\end{equation}
where $\left|x\right>$ denotes a position ...
user224659