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Summary of lecture 22

- Energy levels for a quantum particle in a box

- Energy levels for a quantum linear oscillator

- Energy levels for Bohr’s atom

The Quantum Hydrogen Atom

We shall now revisit the hydrogen atom, an atom containing a nucleus of charge Z and a single electron. We have considered the hydrogen atom before in our discussion of the Bohr model and on energy quantization. That model derived an expression for the quantized energies associated with particular electron orbits. The energy expressions is:

R is called Rydberg constant and expresses the energy is in terms what is required to remove an electron from an atom:

R=2.18x10-18 J

=13.6 eV/molecule (electron volt)

The Quantum Hydrogen Atom

Of course the Bohr model uses a quantization scheme that only yields energies, but not electronic wave functions; it was derived years before wave mechanics was introduced. We shall now reexamine the hydrogenic atom using the Schrodinger equation. The time independent Schroedinger equation for the hydrogen atom is:

Here the potential energy is the Coulombic attraction between the positively charged nucleus and the negatively charged electron

The Quantum Hydrogen Atom

This time-independent Schroedinger equation is never solved in Cartesian coordinates. Like other central force problems, this equation is solved by converting to spherical coordinates:

The Quantum Hydrogen Atom

This time-independent Schroediner equation is never solved in Cartesian coordinates. Like other central force problems, this equation is solved by converting to spherical coordinates:

e-

p+

The Quantum Hydrogen Atom

This time-independent Schroediner equation is never solved in Cartesian coordinates. Like other central force problems, this equation is solved by converting to spherical coordinates:

Once the Schroedinger equation is converted to spherical coordinates, it can be solved by separation of variables by assuming that the wave function is a product of two functions, one of which described the radial component and the other the angular component of the wave function

The Quantum Hydrogen Atom

Like other three dimensional problems, the hydrogen atom wave function is parametrized by three integers n, l, and m that arise out of the solutions of the differential equations that are separated out of the Schroedinger equation

They are called principal, angular and magnetic quantum number. A wave function with a given set of quantum numbers is called an orbital. The first term is called radial wave function, while the second is the angular wave function and is expressed in terms of well-known functions called spherical harmonics

Radial Wave Function

The wave function is the solution of the radial wave equation. The radial function is the dependence of the wave function on the electron-nuclear distance r. The integer n=1, 2, 3 is called the principal quantum number. It is used to calculate the energy:

In order to be correct, one would have to use the reduced mass, but for the hydrogen atom, the reduced mass is the electron mass to within less than 0.1%. The energy is quantized, but only in terms of one integer n derived from the radial equation. This occurs because the potential energy is only dependent on r.

Radial Wave Function

The second integer l is the angular momentum quantum number, which varies from 0 to n-1. As we shall see in a moment, the quantum number l quantizes the total angular momentum L according to the equation:

It is conventional to designate wave functions corresponding to l=0 as s, l=1 as p, and l=2 as d, l=3 as f. s orbitals are easiest to discuss because the electron density is only dependent on r (spherical symmetric).

Radial Wave Function

The radial probability distribution function is the probability that an electron is located in a volume 4pr2 dr. This function is proportional to:

One of several ways to quantify the size of an orbital is to determine the radius that encloses 90% of the total electron density. This is given by the equation:

Radial Wave Function

The best quantitative measure of the size of an orbit valid for all values of n and l is the average distance of an electron from the nucleus:

Note the leading term is just the orbital radius obtained from Bohr theory:

Radial Wave Function

Hydrogen-like orbitals are used to describe the properties of many molecules, for which the Schrodinger equation cannot be solved analytically. It is useful to plot some radial wave functions and probabilities in terms of a0=0.53 A (0.53x10-8 cm) (Bohr radius) and s=r/a0. The radial wave functions have n-1 nodes. For n=1, l=0(s), there are 0 nodes. For n=2, l=0, the node is at r=2a0. For n=3 there are nodes at r=1.9a0 and 7.1a0.

Orbital Shapes

- Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear).

Probability =

Probability is spherical

Angular Wave Function

Y(q,f) is the part of the wave function dependent on the equatorial and azimuthal angles f and q and is called the angular wave function; these are quite famous functions called spherical harmonics. The angular wave function has two more quantum numbers: l and m. These quantum numbers arise because two more physical quantities are quantized. Recall that the principal quantum number n exists because the energy is quantized.

Angular Wave Function

The quantum numbers l and m exist as a result of the quantization of the angular momentum of the electron. The angular momentum is defined as the cross product of the radial vector that defines the classical orbital radius of the electron, the momentum vector

and is thus itself a vector with both direction and magnitude. Both properties are quantized. The l quantum number exists as a result of the quantization of the magnitude of the angular momentum, which is quantized according to the rule:

Angular Wave Function

The final quantum number m exists because the orientation or direction of the angular momentum is also quantized. This can be expressed by saying that the z component of the angular momentum Lz can only assume certain values according to the equation:

The m quantum number varies from –l to +l.

Angular Wave Function

The wave functions Yl,m(q,f) are not easily visualized in any simple coordinate system. Instead, linear combinations of the Yl,m(q,f) wave functions are constructed that are easily visualized in spherical coordinates. For the n=2 p orbitals, these functions are designated Yl,x(q,f),Yl,y(q,f),and Yl,z,(q,f), also called the px, py, and pz orbital wave functions

The p orbital wave functions have a simple sine/cosine dependence on the azimuthal and equatorial angles. For example, the wave function for a 2px orbital has the form:

Angular Wave Function

The form of the angular portion of this function indicates that electron density will be greatest where sinq and cosf are large, i.e. at q=p/2 and f=0. In other words, around the x axis. Similarly, for the py orbital density is greatest around the y axis, and for the pz orbital, density is greatest where q=0, which is near the z axis. Tables of angular wave functions can be found in various texts on quantum mechanics.

Quantum Numbers and Orbitals

n l Orbital ml # of Orb.

0 1s 0 1

0 2s 0 1

1 2p -1, 0, 1 3

0 3s 0 1

1 3p -1, 0, 1 3

2 3d -2, -1, 0, 1, 2 5

Multi-electron Systems

- Can set up the Schrodinger Equation problem for multi-electronic systems
- Can not solve it analytically (basic math principles) as there are too many variables and not enough equations
- Can solve using more complicated methods “Self-consistent field method”
- Result: A series of wave function not unlike that of the one electron system
- Orbitals of multi-electronic systems do not differ significantly from “hydrogenic” wave functions
- Add in a fourth quantum number (magnetic spin)

Angular Wave Function

The form of the angular portion of this function indicates that electron density will be greatest where sinq and cosf are large, i.e. at q=p/2 and f=0. In other words, around the x axis. Similarly, for the py orbital density is greatest around the y axis, and for the pz orbital, density is greatest where q=0, which is near the z axis.

Orbital Shapes

2p (l = 1) orbitals

not spherical, but lobed

labeled with respect to orientation along x, y, and z

Orbital Shapes

3p orbitals

more nodes as compared to 2p (expected)

still can be represented by a “dumbbell” contour

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