All Questions
Tagged with definition hilbert-space
77
questions
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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
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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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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
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votes
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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
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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
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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 ...
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2
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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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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
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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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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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0
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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 ...
- 745
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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?
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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
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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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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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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
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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 "$\...
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3
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529
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$\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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1
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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
$$P_{21}B(1)P^\dagger_{21}|u_i\rangle_1\otimes |u_j\rangle_2=|u_i\rangle_1\otimes |...
- 11.9k
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votes
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answer
443
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What is a quasibound state and how is it different from a bound state?
What is a quasibound state and how is it different from a bound state?
I have read this term in nuclear physics in the context of compound nucleus formation. A compound nucleus $C$ is formed by the ...
- 11.8k
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1
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182
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Requirement of Jordan-Wigner string in creation operator on Fock state
Our lecture notes described the action of the particle creation operator on a fermionic Fock state:
$$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$
I am ...
- 2,604
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What exactly is a Fock state?
I am a bit confused by the way a Fock state is defined and hope to find some clarification.
The Fock space is defined as the direct sum of all $n$-particle Hilbertspaces $H_i$
$$F = H_0 \oplus H_1 \...
- 387
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1
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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
$$\hat\phi(t,\vec{x}) = \sum_n\sum_m\phi^n_m(t,\...
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votes
3
answers
430
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Understanding the bra-ket antilinear correspondence
I can't follow how the above argument leads to (1.8).
I am able to prove it only if I can show $$\langle a | c\rangle+\langle b| c\rangle=(\langle a|+\langle b|)\,|c\rangle$$
But I don't understand ...
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