chapter 7 the hydrogen atom in wave mechanicsphysicsx.pr.erau.edu/courses/coursess2017/ps303/letures...
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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