Quantum Mechanics, Wave Functions and the Hydrogen Atom We have seen how wave functions provide insight into the energy level patterns of atoms and molecules

Quantum Mechanics, Wave Functions and the Hydrogen Atom

• We have seen how wave functions provide insight into the energy level patterns of atoms and molecules. In the one-dimensional, two-dimensional and three- dimensional PIAB models we derive very approximate wave functions and, correspondingly, approximate energies for electrons in atoms and molecules. The PIAB models are useful in that they do “explain” quantization of energy in molecules.

Wave Properties of Small Particles

• The PIAB models also reflect the wave properties of small/light particles. The waves have differing amplitudes (dependent on the values of spatial and time coordinates) and can exhibit nodes. In quantum mechanics the probability of finding a particle at a given point in space is found from the square of the amplitude of the wave function.

Plots of Ψn(x) and Ψn(x)2 for the One-Dimensional PIAB Model

• The graphs on the “next” slide show again the Ψn(x) versus x plots for the one-dimensional PIAB. In all cases the range of x values is defined by 0 ˂ x ˂ L. We note that:

• 1. As the value of the quantum number n increases, the corresponding energy, En, increases.

• 2. As n increases the number of nodes seen for the wave in the box increases.

Plots of Ψn(x) and Ψn(x)2 for the One-Dimensional PIAB Model (continued)

• 3. As n increases the frequency of the wave increases (wavelength decreases). These results should be compared to the familiar result for light where EPhoton = hνPhoton.

• Waves exhibit both positive and negative amplitudes. Nodes are equally spaced.

Particle in Box: Standing Waves, Quantum Particles, and Wave Functions

Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8 Slide 6 of 50

FIGURE 8-20•The standing waves of a particle in a one-dimensional box

ψ, psi, the wave function.Should correspond to a standing

wave within the boundary of the system being described.

Particle in a box.

2

Lsin

nL

x


• The graphs on the “next” slide show the Ψn(x)2 versus x plots for the one-dimensional PIAB. We note that:

• 1. For all n values, Ψn(x)2 is positive. This is expected since Ψn(x)2 gives the probability of finding a particle in a given part of the box. The probability of finding a particle in a particular part of the box can’t be negative.


• 2. In subsequent courses we will see that the area under each of the Ψn(x)2 versus x curves is unity (= 1). This is equivalent to saying that, for the entire box, the probability of finding the particle somewhere is 100%.

• 3. The probability of finding the particle is very different for different x values. We can still identify nodes from the Ψn(x)2 versus x plots. How? Note that the Ψn(x)2 plot for the right half of the box is the mirror image of the plot for the left half of the box.


The probabilities of a particle in a one-dimensional boxFIGURE 8-21

The Hydrogen Atom• The wave functions for the H atom can be

obtained as solutions to the Schrodinger equation, HΨ(r,ϴ,φ) = EΨ(r,ϴ,φ). Aside: This is both an eigenvalue equation and a second order differential equation which will be treated in detail in higher level courses. The use of spherical polar coordinates is often useful when coulombic interactions are important.

The Hydrogen Atom (continued)• The use of spherical polar coordinates also allows

us, in the case of the H atom, to obtain particularly useful solutions to the Schrodinger equation, HΨ(r,ϴ,φ) = EΨ(r,ϴ,φ). The wave functions for the H atom, Ψ(r,ϴ,φ), can be “factored” to give a function which has no angle dependence. This function, R(r), is called the radial wave function. The second part of the total wave function, Y(ϴ,φ), gives the angular dependence of the total wave function.

Wave Functions of the Hydrogen Atom


FIGURE 8-22•The relationship between spherical polar coordinates and Cartesian coordinates

Schrödinger, 1927 Eψ = H ψ

H (x,y,z) or H (r,θ,φ)

ψ(r,θ,φ) = R(r) Y(θ,φ)

R(r) is the radial wave function.

Y(θ,φ) is the angular wave function.

H Atom and Orbitals• As mentioned last day, we need three quantum

numbers to describe the energies of the various energy levels in the H atom. Again, these are n, l and ml. In the H atom each principal energy level or shell is found (by experiment) to consist of a number of subshells which are labelled according to their l values or a letter (s, p, d, f….). In the H atom a particularly simple energy level pattern results.

8-7 Quantum Numbers and Electron Orbitals

• Principle quantum number, n = 1, 2, 3…

• Angular momentum quantum number,l = 0, 1, 2…(n-1)

l = 0, sl = 1, pl = 2, dl = 3, f


Magnetic quantum number, ml= - l …-2, -1, 0, 1, 2…+l

Principal Shells and Subshells


FIGURE 8-23

•Shells and subshells of a hydrogen atom

H Atom and Energy Degeneracy • The previous slide shows that, for the H atom,

several energy levels can have the same energy or are degenerate. For the H atom all of the orbitals making up a subshell have the same energy. As well, the various subshells which comprise a shell are degenerate. This is not the case for multi-electron atoms. However, the pattern of degeneracies seen for the H atom is seen for other monatomic one electron species such as He+ and Li2+.

H Atom Wave Functions. Where are the Electrons?

• For the PIAB model the PIAB wave functions are used to locate nodes and to describe the probability of finding a particle in a particular part of the box. For the H atom this is more difficult. Both one-dimensional and three-dimensional plots are used to provide insight into where electrons are most likely be found when they are “located” in a specific energy level.

Interpreting and Representing the Orbitals of the Hydrogen Atom.

• Represent the probability densities of the orbitals of the hydrogen atom as three dimensional surfaces.

• Each orbital has a distinctive shape.

• Acquire a broad qualitative understanding.


H Atom Wave Functions – Locating Nodes

• For the H atom wave functions there are radial nodes and nodal planes – the latter reflecting the properties of the “angular parts” of the total wave function. We will locate radial nodes for a few orbitals in class and, should time permit, nodal planes as well. The wave functions specific to the 1s, 2s, 3s… orbitals have no explicit angle dependence. Thus, there can only be radial nodes.


H Atom Wave Function Problems• 1. Using the wave functions on the previous

slide determine the number of radial nodes for the H atom 1s, 2s and 3s orbitals.

• 2. Locate the position of the radial node for the 2s orbital of (a) a H atom, (b) a He+ atom and (c) a Li2+ ion.

• 3. Plot R1,0(r) = R(1s) for both the H atom and the He+ ion. What do these plots tell us about the “electron distribution” for the lone electron in the 1s orbital of each species?

Probability Plots for the H Atom

• Chemists often wish to describe the probability of finding the electron in the H atom as a function of its position in three dimensional space. This requires an evaluation of Ψ2 and three dimensional plots. Due to the wave like properties of electrons the maximum value of r that should be used in such plots is not obvious (there is a small likelihood that the electron will be found far from the nucleus).

Orbital Representations

• In practice it is customary to draw a boundary surface enclosing the smallest volume which has, say, a 95% probability of containing the electron. Chemists also speak in using these plots of electron density. The s orbitals are again a special case. The wave functions for s orbitals, the Ψ(r,ϴ,φ), have in this case no angle dependence – the probability of finding the electron somewhere in space depends “only” on the r value.

s orbitals


FIGURE 8-24

•Three representations of the electron probability density for the 1s orbital

2s orbitals


FIGURE 8-24

•Three-dimensional representations of the 95% electron probability density for the 1s, 2s and 3s orbitals

Test 1 Examples

• 1. (a) Write the balanced chemical equation corresponding to the molar enthalpy of formation of copper (II) nitrate hexahydarate, Cu(NO3)2

.6H2O(s).

• (b) Write the balanced chemical equation for the complete combustion of benzoic acid, C6H5COOH(s).

Test 1 Examples

• 2. The atmospheric pressure at the summit of Mt Everest is 28.9 kPa and the air density is 0.436 kg/m3. Determine the air temperature assuming an effective molar mass for air of 29.2 g/mol.

Test 1 Examples

• 7. A gas expands at constant temperature from a volume of 3.50 L to 7.60 L. Find the work done by the gas if it expands (a) against a vacuum and (b) against a constant external pressure of 1.45 atm.

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Quantum Mechanics, Wave Functions and the Hydrogen Atom We have seen how wave functions provide insight into the energy level patterns of atoms and molecules