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-...
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  • 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 ...
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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 ...
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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 ...
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  • 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 ...
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3 votes
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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 ...
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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 $$ H:=\sum_{n\ge0}E_n|n\rangle\langle n| $$ ...
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1 vote
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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. ...
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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 ...
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  • 21
1 vote
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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(...
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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 ...
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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 ...
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  • 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 ...
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