Quantum Physics: Energy Levels Of Hydrogen Atoms

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Quantum Physics Exam 3

Problem1. Hydrogen Essentials

PART A: Make a plot of the energy levels of the hydrogen atom. Plot the energy values En in the vertical direction for n = 1, 2,3,4,5. Plot the orbital angular momentum quantum number in the horizontal direction for l = 0,1,2,3,4. For each n, show every allowed value of l. Label every energy level spectroscopically (1s, 2s, 2p, ...). Indicate the m degeneracy of each l level. Show that the total degeneracy of each En is n2.

Solution:

Bohr radius separates the potential energy of two electrons according to the equation: U=e2/4πε0a0 …Equation [1]. This equation represents one atomic unit of energy.

The Schrödinger equation for the Hydrogen

The levels(n=1,2,3,4) are represented in black.

How the energy levels are lined up. Explanation:

From the sketch in Figure 1below, for level n=1, there is only one value allowed for l. The value is l=0. the potential well that corresponds to l=1(light blue) is above n = 1. For n =2, there are two values allowed for l. The possible values of l are l=0 and l=1because this energy level is above the wells of l=0 and l=1and below l=2 (green) and l=3 (violet). In other words, there is only one possible orbital (1s) for the first energy level (n=1)). The existence of other superior orbitals (1p,1d,etc) is forbidden by the fact that their energy wells (in which they lie) are above the n=1n=1 energy. Figure 1: Sketch where n = 1

PART C: Sketch just the l = 0 effective potential. Add the locations of the n = 1,2,3 energy levels to your sketch. Sketch the radial wave functions for each of these energy levels and label each wave function with the appropriate Rnl designation. Sketch the corresponding probability distributions per unit volume | (r)|2 dV. Sketch the corresponding radial probability distributions 4⇡r2 | (r) |2

From Equation [10], the normalized time-dependent wave function at a later time, t=to can be expressed as follows:

{|ψ(x,t=to)}=〖3/√23 ∑▒e〗^(-(iEnt)/h) |ψ1 (x)-〖1/√23 ∑▒e〗^(-(iEnt)/h) |ψ2 (x)+ 〖2/√23 ∑▒e〗^(-(iEnt)/h) |ψ3 (x)- 〖3/√23 ∑▒e〗^(-(iEnt)/h) |ψ3 (x) … Equation [12]

PART D: If you measure the energy at time t, what are the possibilities and what are the probabilities?