Chapter 7

The Hydrogen Atom in Wave Mechanics One dimensional Schrodinger Equation

− ℏ$

%&'

$(')$

− *$

+,-.)𝜓 𝑥 = 𝐸𝜓 𝑥

Solutions are: 𝜓3 𝑥 = 𝐴𝑥𝑒6)/8. = %

8.9/$ 𝑥𝑒6)/8. (ground state)

Solutions to the Three Dimensional Schrodinger Equation (Hydrogen Atom):

−ℏ%

2𝑚𝜕%𝜓𝜕𝑥% +

𝜕%𝜓𝜕𝑦% +

𝜕%𝜓𝜕𝑧% + 𝑈 𝑥, 𝑦, 𝑧 𝜓 𝑥, 𝑦, 𝑧 = 𝐸𝜓 𝑥, 𝑦, 𝑧

The potential energy term:

𝑈 𝑥, 𝑦, 𝑧 = 𝑈 𝑟, 𝜃, 𝜙 = 𝑈 𝑟 = −𝑒%

4𝜋𝜖H1𝑟

The Schrödinger Equation becomes: −ℏ%

2𝑚𝜕%𝜓𝜕𝑟% +

2𝑟𝜕𝜓𝜕𝑟 +

1𝑟% sin 𝜃

𝜕𝜕𝜃 sin 𝜃

𝜕𝜓𝜕𝜃 +

1𝑟% sin% 𝜃

𝜕%𝜓𝜕%𝜙 −

𝑒%

4𝜋𝜖H1𝑟 𝜓 𝑟, 𝜃𝜙 = 𝐸𝜓(𝑟, 𝜃, 𝜙)

Solve this differential equation using separation of variables

𝜓 𝑟, 𝜃, 𝜙 = 𝑅 𝑟 𝛩 𝜃 𝛷(𝜙) This results in three separate equations. The radial equation becomes:

−ℏ%

2𝑚𝑑%𝑅𝑑𝑟% +

2𝑟𝑑𝑅𝑑𝑟 + −

𝑒%

4𝜋𝜖H1𝑟 +

ℓ ℓ + 1 ℏ%

2𝑚𝑟% 𝑅 𝑟 = 𝐸𝑅(𝑟)

𝑅Tℓ =the Associated Laguerre polynomials

where ℓ = 0, 1, 2, … and ℓ ≤ 𝑛 − 1 𝛩ℓ&ℓ 𝜃 =the Associated Legendre polynomials

where −ℓ... ≤ 𝑚ℓ ≤ ...+ℓ

𝛷&ℓ 𝜙 = 12𝜋

𝑒±[&ℓ\

Radial Wave Functions

Angular Momentum States ℓ = 2 Length of the angular momentum vector:

𝐿 = ℓ(ℓ + 1)ℏ

𝐿^ = 𝑚ℓℏ Degenerate Energy States

Relationship between the magnetic dipole moment and angular momentum Angular Momentum:

• Orbital Angular Momentum 𝐿 • Spin Angular Momentum 𝑆

𝜇a = 𝐼𝐴 =

𝑞2𝜋𝑟𝑚/𝑝(𝜋𝑟

%)

𝜇a =

e%&𝐿 𝜇a = −

*%&𝐿

𝜇a^ = −

*%&𝐿^ 𝐿^ = 𝑚ℓℏ

Forces on an electric dipole

Forces on a magnetic dipole

The Stern-Gerlach Experiment (Observing spatial quantization)

The number of angular momentum sub-orbitals The number of 𝑚ℓstates = 2ℓ + 1 However, in some cases, the number of discrete images on the screen did not agree with this expectation (i.e., an odd number). Sometimes there were an even number of states (e.g., ℓ = 3

%

Answer →Spin Angular Momentum !! Spin Angular Momentum Length of the spin vector

𝑆 = 𝑠(𝑠 + 1)ℏ Measured states of the spin vector

𝑆^ = 𝑚hℏ

Magnetic Dipole Moment due to Spin of the Electron

𝜇 = −𝑔*𝜇j 𝑆ℏ

where 𝑔* = 2.0023193043617(15) measured to 12 sig. digits Energy Levels (Grotrian diagrams) Hydrogen Atom (with no magnetic field) Selection Rules

𝛥ℓ = ±1

The Zeeman Effect Potential energy function 𝑈 = −𝜇a ⋅ 𝐵 Normal Zeeman Effect Additional Selection Rule: 𝛥𝑚ℓ = 0, ±1 along with 𝛥ℓ = ±1

Fine Structure (Spin-Orbit Interactions)

𝑈 = −𝜇s ⋅ 𝐵