All Questions
8
questions
0
votes
1
answer
79
views
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
1
vote
1
answer
513
views
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
1
vote
1
answer
441
views
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
3
votes
1
answer
189
views
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
0
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 ...
- 773
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
42
votes
5
answers
8k
views
Hilbert space vs. Projective Hilbert space
Hilbert space and rays:
In a very general sense, we say that quantum states of a quantum mechanical system correspond to rays in the Hilbert space $\mathcal{H}$, such that for any $c∈ℂ$ the state $\...
- 4,685