1. CHAPTER 7 The Hydrogen Atom
Orbital Angular momentum
Application of the Schrödinger Equation to the Hydrogen Atom
Solution of the Schrödinger Equation for Hydrogen
Werner Heisenberg
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The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions. All its qualities are inferential; no material properties can be directly attributed to it. An understanding of the atomic world in that primary sensuous fashion…is impossible.
- Werner Heisenberg

2. Orbital Angular Momentum Quantum Number ℓ
Classically, angular momentum is . But quantum-mechanically, the total orbital angular momentum L is an operator, e.g.,
The eigenvalues of are functions of ℓ:
So, in an ℓ = 0 state:
This disagrees with Bohr’s semi- classical planetary model of electrons orbiting a nucleus, for which L = nħ, where n = 1, 2, …
Classical orbits—which don’t exist in quantum mechanics

3. Orbital Magnetic Quantum Number mℓ
Possible total angular momentum directions when ℓ = 2
mℓ is an integer. It determines the z component of L:
Example: ℓ = 2:
Only certain orientations of are possible.
And (except when ℓ = 0) we just don’t (and cannot!) know Lx and Ly!

4. Application of the Schrödinger Equation to the Hydrogen Atom
The potential energy of the electron-proton system is electrostatic:
For Hydrogen-like atoms (He+ or Li++), replace e2 with Ze2 (where Z is the atomic number).
The three-dimensional time-dependent Schrödinger Equation:
where:
which becomes the three-dimensional time-independent Schrödinger Equation:
where:

5. Spherical Coordinates
The potential (central force) V(r) depends only on the distance r between the proton and electron, where
As a result of this radial symmetry, we must transform to spherical coordinates:
Polar angle
The inverse relations:
Azimuthal angle

6. The Schrödinger Equation in Spherical Coordinates
Transformed into spherical coordinates, the Schrödinger equation becomes:
Also, use the reduced mass m instead of the electron mass me

7. Separable Solution
The wave function y is a function of r, q, f. This is a potentially complicated function. Assume optimistically that y is separable, that is, a product of three functions, each of one variable only:
This would make life much simpler—and it turns out to work!

8. Solution of the Schrödinger Equation for Hydrogen
Start with Schrodinger’s Equation:
Substitute:
Multiply both sides by -r2 sin2q / R f g:

9. Solution of the Schrödinger Equation for Hydrogen
r and q appear only on the left side and f appears only on the right side. The left side of the equation cannot change as f changes. The right side cannot change with either r or q. Each side needs to be equal to a constant for the equation to be true. Set the constant to be −mℓ2
Azimuthal-angle equation
Sines and cosines satisfy this equation, but it’s convenient to choose a complex exponential:
g(f) = exp(imℓf)

10. Quantization of mℓ g(f) = exp(imℓf)
exp(imℓf) satisfies the azimuthal equation for any value of mℓ. But a type of boundary condition exists because f and f + 2p are actually the same angle, so:
So:
Canceling:
mℓ must be an integer (positive or negative) for this to be true.

11. Solution of the Schrödinger Equation for H
Now replace by −mℓ2:
Divide by sin2q and rearrange:
Now, the left side depends only on r, and the right side depends only on q. We can use the same trick again!
Set each side equal to the constant ℓ(ℓ + 1).

12. Separation of the Schrödinger Equation for H
The Associated Laguerre equation
Set each side equal to the constant ℓ(ℓ + 1).
Radial equation
Polar-angle equation
We’ve separated the Schrödinger equation into three ordinary second-order differential equations, each containing only one variable.

13. Solution of the Radial Equation for H
The radial equation is called the associated Laguerre equation and the solutions R are called associated Laguerre functions. There are infinitely many of them, for values of n = 1, 2, 3, … Assume that the ground state has n = 1 and ℓ = 0. Let’s find this solution. The radial equation becomes: The derivative of yields two terms:

14. Solution of the Radial Equation for H
Try a solution A is a normalization constant. a0 is a constant with the dimension of length. Take derivatives of R and insert them into the radial equation.
To satisfy this equation for any r, both expressions in parentheses must be zero.
Set the second expression equal to zero and solve for a0:
Set the first expression equal to zero and solve for E:
Both are equal to the Bohr results!

15. Hydrogen-Atom Radial Wave Functions
The solutions R to the radial equation are called Associated Laguerre functions. There are infinitely many of them, for values of n = 1, 2, 3, … and ℓ < n.

16. The Polar-Angle Equation
Note the presence of mℓ. Recall that the solutions to the azimuthal equation, exp(imℓf), also involve mℓ.
So solutions to the polar- and azimuthal-angle equations are linked.
We usually group these solutions together into functions called Spherical Harmonics:
Spherical harmonics
Solutions to the polar-angle equation above involve sums and products of sines and cosines.

17. Normalized Spherical Harmonics
Spherical harmonics are the solutions to the combination polar- and azimuthal-angle equations.

18. Complete Solution of the Radial, Angular, and Azimuthal Equations
The total wave function is the product of the radial wave function Rnℓ and the spherical harmonics Yℓm and so depends on n, ℓ, and mℓ. The wave function becomes: ℓ
where only certain values of n, ℓ, and mℓ are allowed.