All Questions
Tagged with definition hilbert-space
77
questions
0
votes
1
answer
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
1
vote
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
0
votes
1
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58
views
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
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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 ...
- 77
2
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
3
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 $\...
- 493
2
votes
1
answer
87
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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
0
votes
1
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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 ...
- 773
6
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 ...
- 155
2
votes
3
answers
723
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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
8
votes
2
answers
677
views
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