Question

In several bound systems, the quantum-mechanically allowed energies depend on a single quantum number we found in section 5.5 that the energy levels in an infinite well are given by, ${\mathit{E}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{1}}{\mathit{n}}^{\mathbf{2}}$where$\mathit{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{.....}$andis a constant. (Actually, we known what${\mathit{a}}_{\mathbf{1}}$is but it would only distract us here.) section 5.7 showed that for a harmonic oscillator, they are${\mathit{E}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{2}}\mathbf{(}\mathbf{n}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{)}$, where$\mathit{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{.....}$(using an$\mathit{n}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}$with n strictly positive is equivalent towith n non negative.) finally, for a hydrogen atom, a bound system that we study in chapter 7,${\mathit{E}}_{\mathbf{n}}\mathbf{=}\mathbf{-}\frac{{\mathbf{a}}_{\mathbf{3}}}{{\mathbf{n}}^{\mathbf{2}}}$, where$\mathit{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{.....}$consider particles making downwards transition between the quantized energy levels, each transition producing a photon, for each of these three systems, is there a minimum photon wavelength? A maximum ? it might be helpful to make sketches of the relative heights of the energy levels in each case.

Short answer

InfiniteWells: Maximum photon wavelength exists

Harmonic Oscillator:Maximum photon wavelength exists

HydrogenAtom: Minimum photon wavelength exists

Step-by-step solution

Given information.

System 1:

Consider the formula for infinite well energy levels:

${\mathbf{E}}_n\mathbf{=}{\mathbf{a}}_1{\mathbf{n}}^2$

Here,$n=1,2,3\dots$and$a_1$is a constant.

System 2:

Consider harmonic oscillator energy levels:

${\mathbf{E}}_n\mathbf{=}{\mathbf{a}}_2\mathbf{(}\mathbf{n}\mathbf{-}\frac{1}{2}\mathbf{)}$

Here, $n=1,2,3\dots$

System 3:

Consider hydrogen atom energy levels:

${\mathbf{E}}_n\mathbf{=}\frac{\mathbf{-}{\mathbf{a}}_3}{{\mathbf{n}}^2}$

Here,$n=1,2,3\dots$

It is to be noted that particles are making downward transitions between the quantized energy levels, and each transition thatproduces a photon.

Explain System 1.

The maximum photons well comprises of the finite photon wavelength. For the infinite possible energy levels there do not exist any maximum photon energy or any photon wavelength of minimum value. Though the minimum energy difference arises between the different energy levels, while adding the energy level will increase the separation between them. Consider the closest energy levels are$n=1,2$along with the shortest separation distance producing the photon with maximum wavelength.

Explain System 2.

The maximum photon wavelength comes for the harmonic oscillator as it was in the case of infinite well where there were infinite number of energy levels with no minimum wavelength. The second system has the evenly spaced energy levels with the maximum wavelengths corresponding to the transitions between the two energy levels.

Explain System 3. 

Note the hydrogen atoms has minimum photon wavelength while the atom has the maximum energy of zero with the largest energy transition from 0 energy state to the lowest negative energy state and thus creating the minimum wavelength of photon. In the same way there is no maximum wavelength as the energy levels is becomes closer as the value of n increases and the tradition energy decreases to zero.

Step 5:Determine the final answer

InfiniteWells: Maximum photon wavelength exists

Harmonic Oscillator:Maximum photon wavelength exists

HydrogenAtom: Minimum photon wavelength exists