16
votes
Does the "particle in a box" necessarily form a standing wave?
The particle in a box does not necessarily form a standing wave. In fact in quantum mechanics these are states of definite energy that have trivial time evolution, but superpositions of them have non-...
- 35k
13
votes
Accepted
What is the correct separable Schrödinger equation in spherical coordinates?
Both formulas are correct. And it is essentially a matter of taste
which one you prefer:
If you use the separation approach
$$\Psi(r,\theta,\phi) = R(r) \cdot \Theta(\theta) \cdot \Phi(\phi)$$
then ...
- 26.8k
5
votes
Does the "particle in a box" necessarily form a standing wave?
It’s important to make a distinction between two different “particle in a box” setups.
1.) Infinite Potential Well
Imagine that the “box” is a region with $0$ potential energy, and everywhere outside ...
- 378
4
votes
Does the "particle in a box" necessarily form a standing wave?
It is completely possible for a particle in a box not to be in a standing wave state. In fact there are infinitely many more non-standing wave solutions to the particle in the box than there are ...
- 4,017
3
votes
Can energy levels rise faster than $n^2$?
In this answer we generalize OP's 1D semiclassical WKB estimate for a power law potential $\Phi(x) \propto |x|^{\alpha}$ to an arbitrary potential $\Phi(x)$. If $\ell(V)$ denotes the accessible length ...
- 170k
3
votes
Accepted
Separability of Hamiltonian and Factorization of Wavefunction
I think your interpretation of Shankar's statement is too strong and that cannot be true. Shankar's statement is that separable states make up a complete eigenbasis of the Hamiltonian $H$. Your ...
- 116
1
vote
Can energy levels rise faster than $n^2$?
It sort of depends on what restrictions you impose on $H$.
If we let $|n\rangle$ be the eigenstates of the harmonic oscillator, then you can surely define
$$
H:=\sum_{n\ge0}E_n|n\rangle\langle n|
$$
...
1
vote
Accepted
Deriving time-independent Schrodinger equation for $n$ qubit system with a constant Hamiltonian
Yes, $f(t)$ is usually called the time evolution operator, or propagator of the problem. The conventional notation for $f(t)$ is $U(t)$, which comes from the fact that $U(t)$ is a unitary operator.
...
- 2,903
1
vote
What is the correct separable Schrödinger equation in spherical coordinates?
If I am not wrong, then the second one is used for (I guess) when we solve the radial part. We have to put $R(r)=u(r)/r$, so maybe in second one they have written $R(r)/r$ instead of $R(r)$ is because ...
- 21
1
vote
Accepted
Interpreting the Time Evolution Operator on a hands-on example
It may help to look at a simple example. If your state vector is $\pmatrix{1\\0}$ at $t=0$, then you will have
$$|\psi(t)\rangle = \pmatrix{\cos(\omega t)\\-i\sin(\omega t)}$$
In particular,
$$|\psi(...
- 51.8k
1
vote
Interpreting the Time Evolution Operator on a hands-on example
It is indeed good your intuition that the time evolution is a rotation but to get the general sense of what is happening you have to think in terms of Hilbert spaces and basis on such spaces.
What is ...
1
vote
Using the uncertainty principle to estimate energies in ground states
Building on the last equation in Semoi's answer:
\begin{equation}
\left\langle E \right\rangle = \frac{(\Delta p)^2}{2m} + \frac{m\omega^2 (\Delta x)^2}{2},
\tag{1}
\end{equation}
if you want to ...
- 135
1
vote
Can we solve the particle in an infinite well in QM using creation and annihilation operators?
One has to distinguish creation/annihilation operators and raising/lowering operators.
Second quantization is the method for introducing creation and annihilation operators for any system. This is ...
- 39.1k
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