23 questions linked to/from Discreteness of set of energy eigenvalues
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### Trying to understand why energy must be quantized/discrete [duplicate]

So I'm reading about bound systems right now in my quantum text. It is beginning to explain why energy must be quantized, and is doing so by introducing the reader to the one dimensional "quanton in a ...
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### What is the cause of discrete or quantized energy levels in an atom? [duplicate]

I understand how it is that electrons move from one energy state to another, however I've not been able to find anywhere that describes why an atom has any particular states. Why should an atom of ...
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### Why is the set of physically acceptable solutions to the time-independent Schrodinger equation discrete? [duplicate]

The time-independent Schrodinger equation for a particle moving along the x-axis is: $$H\psi (x) + V(x) \psi (x) = E \psi(x)$$ I know that the physically acceptable solutions to this equation can ...
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### Why the eigenvalues of hamiltonian are discrete in bounded systems (only discrete energy levels)? [duplicate]

In one dimensional motion the general potential is given as in the figure above When energy is between V min and V 1 the energy levels are discrete like in the potential barrier and well example or ...
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### Do non-vanishing boundaries conditions necessarily force a discrete/quantized set of solutions for the Schrödinger Equation? [duplicate]

I was reviewing past exams and I found a question where I could give no satisfactory answer. The core of the question is the following: If there are boundary conditions, does this necessarily imply, ...
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### Reason for the discreteness arising in quantum mechanics?

What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
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### Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?

When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
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### Bounded and Unbounded (Scattering) States in Quantum Mechanics

I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state ...
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### Why do electrons occupy in discrete energy states?

Why can't there be any continuous energy band in an atom?
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### How to prove bound state's spectrum must be discrete and scattering state's spectrum must be continuous?

Consider $d$-dim Schrodinger equation without internal degree of freedom, that is, we don't consider spin etc. How to prove that bound state's spectrum must be discrete and scattering state's ...
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### When is energy discrete/quantized for a potential well?

Specifically, my question is: Should one expect energy quantization for a particle in the following potential well? More generally, how can one tell whether or not energy should be discrete/...
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### Energy quantization

Why does a free particle give rise to a continuous spectrum of energy eigenvalues while a particle in a bound potential gives rise to a discrete spectrum?
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### How does quantum mechanics and quantum field theory explains discrete energy levels of particles?

Please give me a brief explanation as to how qm and qft describe and explain the energy level that exist in an atom. I understand that in QM energy state is quantized but does not offer an ...
My question is a duplicate of this. Please consider the equation $\nabla^2\psi + (2m/\hbar^2)[E-V]\psi=0$ (1) Potential of electron revolving hydrogen atom is given as \$V=\frac{-e}{4\pi\...