Physical Chemistry Third Edition

17.6 The Time-Dependent Wave Functions of the Hydrogen Atom 753

Exercise 17.14

Sketch rough graphs of the radial distribution function for the following:

a.R 43

b.R 75

c.R 50

PROBLEMS

Section 17.5: Expectation Values in the Hydrogen
Atom

17.35Calculate〈r〉, the expectation value of the distance from
the nucleus for the 1sstate of a hydrogen-like atom for
Z1,Z2, andZ3.

17.36a.Calculate the expectation value〈r〉for a hydrogen-like
atom in the 1sstate. Why is this not equal to〈 1 /r〉−^1?

b.Calculate〈r^2 〉for a hydrogen-like atom in the 1sstate.

Why is this not equal to〈r〉^2?

c.Find the most probable value ofrfor a hydrogen-like

atom in the 1sstate. Why is this not equal to〈r〉?

17.37Find the most probable distance of the electron from the

nucleus for a hydrogen atom in the 211 (2p1) state.

17.6 The Time-Dependent Wave Functions of the Hydrogen Atom

We can write a time-dependent hydrogen atom wave function from our orbitals, using

Eq. (15.2-18):

Ψnlm(r,θ,φ,t)ψnlm(r,θ,φ)e−iEnt/ ̄h (17.6-1)

The probability densityΨ∗Ψis time-independent since the complex exponential and

its complex conjugate cancel each other.

|Ψnlm(r,θ,φ,t)|^2 ψ∗nlm(r,θ,φ)eiEnt/h ̄ψnlm(r,θ,φ)e−iEnt/h ̄ψ∗ψ (17.6-2)

Our hydrogen atom orbitals correspond to stationary states, as do all wave functions

that are products of a coordinate factor and a time factor. The expectation value of

any time-independent variable in a stationary state is time-independent, and can be

calculated from the coordinate wave function, as shown in Eq. (16.4-4).

The real orbitals correspond to standing waves, with stationary nodes. The complex

orbitals withm0 do not represent standing waves, even though they correspond to

stationary states. TheΦmfactor gives the following complex exponential factor in the

time-dependent wave function:

exp(imφ−iE t /h ̄)cos(mφ−Et/h ̄)+isin(mφ−Et/h ̄) (17.6-3)

Form0, this corresponds to a motion of the nodal planes of the real and imaginary

parts around thezaxis, and constitutes a traveling wave moving around the nucleus.

It is easy to visualize an electron orbiting around thezaxis like a classical particle,

producing the angular momentum values that we have obtained. However, if we take

Eas negative, the motion is clockwise form>0 and counterclockwise form< 0

when viewed from the positive end of thezaxis. This direction of motion is opposite