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### How can theories about 1D or 2D systems be generalized for 3D systems?

I was watching a lecture video from MITx $^\dagger$ by professor Barton Zwiebach. He proved a pretty cool theorem "every attractive 1-dimensional potential has a bound state"; however, that only holds ...
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### Question on Uncertainty Principle in a Potential Well

The following is the problem that I'm struggling with from the book of Gasiorowicz: If we have an electron in a potential well of width $a$ and depth $V_0$, then the kinetic energy is, by the ...
From Griffiths, Introduction to Quantum Mechanics, pg. 73: Evidently, the delta-function well, regardless of its "strength" $\alpha$, has exactly one bound state $$\psi(x) = \frac{\sqrt{m \... 2answers 1k views ### How to predict bound states in a 1D triangular well? Assume we have a (single) particle in a potential well of the following shape: For x \leq 0, V = \infty (Region I) For x \geq L, V = 0 (Region III) For the interval x > 0 to x < L,... 1answer 134 views ### Bound state in potential less 0 How to prove that there is a bound state in the potential U(x) = -A e^{-a |x|}, where for all a \in \mathbb{R} and A>0. I heard that we can say something to the minimum of this form  \left( \... 3answers 1k views ### Bound states of the V(x)=\pm \delta'^{(n)}(x) potential? The \delta(x) Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives \delta', \delta'', ... exist also, do they? If yes, can we look for ... 0answers 163 views ### Bound states in 1D & 2D [duplicate] Why does Mother Nature allow bound states in arbitrarily weak attractive potential in 2D but not in 3D? See, for example, this article, arXiv:math-ph/0208011. 0answers 231 views ### How to compute minimum shallowness of quantum well to have at least one bound state? Given a potential V, how does one compute how shallow the potential can be such that it allows at least one bound state? 1answer 1k views ### Am I missing a trick to solving a 3D potential well problem? I was playing around with a 3-D potential V such that V_{(r)} = 0 for r<a, and V_{(r)} = V_0>0 otherwise. By using the Schrödinger Equation, I showed that:$$\frac{-\hbar}{2m}\frac{1}{r^...
What will happen if the potential is less than 0, for instance $V(x)=-10eV$. Is this means there will be no bound states? Since solution to the time independent Schrodinger equation (those discrete ...