All Questions
Tagged with normalizationhilbert-space
150 questions
1vote
1answer
106views
Question on the square-integrability of the given wavefunction at origin and infinity
I have this function as a wavefunction of a quantum system: $$\psi(r)=N r^a \exp\left(br^2 + cr+\frac d{r^3}+\frac e{r^2}+\frac f{r}\right)$$ where $r$ is the radial parameter ranging on the interval $...
- 13
-2votes
1answer
45views
Quantum scattering states [duplicate]
While studying about scattering states in quantum mechanics we come up with terms like Transmission coefficient and Reflection coefficient in consequence of Obtaining two solutions for x<0 and x>...
-1votes
1answer
87views
What function should be used to create wave packet for a particle moving in a constant potential if the energy of the particle is greater than $V_0$? [closed]
I have been trying to solve for a particle under a constant potential $V_0$. Now the the energy states are plane waves and to form a normalizable position state we need to form a superposition of all ...
3votes
1answer
203views
In perturbation theory, are there two or three summation terms in the second-order correction to the eigenfunctions?
Context This question is a narrow one and it is specifically related to non-degenerte, time-independent perturbation theory. In working through [1], Sakurai offers in Eq. (5.1.44) that the second-...
- 734
6votes
3answers
534views
Can we no longer predict the behavior of a particle with a definite position?
This might be a really dumb question as I am just learning QM for the first time. Shankar says that physically interesting wavefunctions can be normalized to a unit $L^2$-norm: $$\int_{-\infty}^{\...
1vote
1answer
144views
Dirac-Delta from Normalization of Continuous Eigenfunctions
I'm following this paper, and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy \begin{equation} \langle f_s|f_{s'}\rangle = \int_{0}^{1} \...
0votes
2answers
110views
Why do we write in general certain eigenfunctions with constants when the weights depend on the hamiltonian?
As a matter of habit, I've simply written out eigenfunctions of spin systems, say, with the usual normalization constants as weights, but now I'm being asked to write them in terms of Hamiltonian ...
0votes
0answers
50views
About momentum states covariant normalization
I'm following QFT of Schwrtz and I have a doubt about Eq. (2.72). In particular, from Eq. (2.69): $$[a_k,a_p^\dagger]=(2\pi)^3\delta^3(\vec{p}-\vec{k}),\tag{2.69}$$ and Eq. (2.70): $$a_p^\dagger|0\...
- 147
1vote
1answer
131views
Confusion on Shankar's Motivation for the Dirac delta Function
I was reading Shankar's Principles of Quantum Mechanics and got confused on page 60, where he motivates the delta function from the normalization problem of the inner product for function spaces. We ...
- 13
2votes
1answer
88views
Square Integrability of spherical symmetric wave
In my class we were discussing some wave equation for a spherical symmetric wave $u(t,r)$ and my professor investigated the behaviour of the solutions asymptotics $r \rightarrow \infty$. The solution ...
- 813
-3votes
1answer
90views
Does the inner product of wavefunctions really have units? [closed]
Let $\psi(x)$ and $\phi(x)$ be wavefunctions. I usually see the inner product defined as $$\int dx\, \overline{\psi(x)} \, \phi(x)$$ and interpreted, I think, as "the amplitude that state $\phi$ ...
- 120
0votes
0answers
82views
Angular momentum completeness relation
Can anyone tell me if the angular momentum completeness relation is given by $$ \sum_{l=0}^{\infty} \sum_{m=-l}^{l} (2l + 1) |l,m\rangle\langle l,m| = I $$ or $$ \sum_{l=0}^{\infty} \sum_{m=-l}^{l} |l,...
- 1,552
0votes
0answers
106views
Continuum bases: why do we use dirac delta function? [duplicate]
In discrete bais, we can express a vector as $$ |\psi\rangle=\sum_{i} c_i|e_i\rangle $$ with orthonormality $$ \langle e_i|e_j\rangle=\delta_{ij}.$$ $\delta_{ij}$ is usual kronecker delta. If we ...
0votes
1answer
95views
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 ...
- 834
0votes
2answers
189views
Normalization to unity, Projection Operators in QM
I have a question about something that is stated in Sakurai's MQM. It's written that if one runs a sequence of selective measurements (namely, a sequence of independent stern-gerlach apparatuses) ...
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