# When a particle in one dimension have discrete spectrum?

• fab333
In summary, the conditions for a system to have discrete energy levels include having a potential that is finite and has a minimum value. In the case of a one-dimensional system with the potential V(x)=b|x|, the spectrum is discrete. To prove this, the energy eigenvalues must satisfy the condition given by the book, which involves calculating a definite integral. This is due to the fact that when the minimum potential is greater than the energy, the solutions become complex and the energy takes on a continuous form. However, when the energy is greater than the minimum potential, the solutions are discrete, indicating a bound particle.
fab333
What are the conditions for which it can be concluded that a system has discrete energy levels?
For example a system in one dimension with the potential
$V(x)=b|x|$
has only a discrete spectrum. How I can prove it?
My book says moreover that the energy eigenvalues have to satisfy the condition
$\lmoustache_{x_1}^{x_2} dx \sqrt{2m[E- \lambda |x|]} = (n+1/2) \pi \hbar$
why?

thanks for help.

In general, in cases where the minimum potential is bigger than the energy of the particle, you find that the general solution will be complex, and the energy takes a continuous form. on the other hand when the energy is bigger than the minimum potential, the solutions take discrete values as we have a bound particle.

## 1. What is a discrete spectrum in one dimension?

A discrete spectrum in one dimension refers to the set of distinct energy levels that a particle can occupy in a one-dimensional system. This means that the particle's energy can only take on specific, quantized values rather than a continuous range.

## 2. How is a discrete spectrum different from a continuous spectrum?

A continuous spectrum refers to a system in which the particle can take on any value of energy within a certain range, while a discrete spectrum is limited to specific, quantized energy levels. In other words, a continuous spectrum has an infinite number of possible energy values, while a discrete spectrum has a finite number of energy values.

## 3. What causes a particle in one dimension to have a discrete spectrum?

A particle in one dimension can have a discrete spectrum when it is confined to a small region or bound within a potential well. This confinement leads to the quantization of the particle's energy, resulting in a discrete spectrum.

## 4. Can a particle in one dimension have both a discrete and continuous spectrum?

Yes, it is possible for a particle in one dimension to have both a discrete and continuous spectrum. This can occur when the particle is confined to a potential well with a finite depth, resulting in a discrete energy spectrum for bound states and a continuous energy spectrum for unbound states.

## 5. How is a discrete spectrum experimentally observed?

A discrete spectrum can be experimentally observed through spectroscopy techniques, such as absorption or emission spectroscopy, which involve shining light onto a system and measuring the wavelengths of light absorbed or emitted. The presence of specific, discrete energy levels in the system can be identified through the resulting spectral lines in the measured data.

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