All Questions
Tagged with definition hilbert-space
77
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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^...
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vote
1
answer
72
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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. ...
- 1,619
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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 ...
- 342
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112
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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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1
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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 ...
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4
answers
501
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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 ...
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1
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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
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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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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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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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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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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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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