Eigen Value and Function of L | Lecture Note - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

36

Pages:

5
Author:

Jabin F.

Eigen value and eigen function of Lz: Let ψ(r, θ, φ) = Φ(φ) be eigen fn of Lz with eigen value λ. L̂z Φ(φ) = λ Φ(φ) -iħ ∂/∂φ Φ(φ) = λ Φ(φ) => L̂z² Φ(φ) = λ² Φ(φ) => -ħ² ∂²/∂φ² Φ(φ) = λ² Φ(φ) d²Φ(φ)/dφ² + λ²/ħ² * Φ(φ) = 0 m² + λ²/ħ² = 0 m = ± λi/ħ Φ(φ) = Ae^(iλφ/ħ) + Be^(-iλφ/ħ) let m = ± λ/ħ => λ = ± mħ Φ(φ) = A'e^(±iλφ/ħ) = A'e^(imφ) Φ(φ) = A'e^(imφ) Condition: Φ(φ) = Φ(2π + φ) Equation: A'e^(imφ) = A'e^(im(2π+φ)) Derived equation: e^(im2π) = 1 Solution: e^(±im2π) = e^(±i2πn) m = ±n, m = ±1, ±2, ±3, ±4 ... m = 0, 1, 2, 3, 4... Φ(φ) = A'e^(imφ) λ = mħ Normalization cond'n |A'|² ∫(0 to 2π) dφ = 1 |A'|² x 2π = 1 |A'|² = 1 / 2π A' = 1 / √(2π) Normalized eigen function of Lz Φm(φ) = (1/√(2π)) e^(imφ) m = 0, ±1, ±2, ±3, ... Eigen value of Lz, λ = mħ m = 0, ±1, ±2, ±3, ... Lz Φm(φ) = mħΦ(φ) |m, l, m> → Ylm(θ, φ) = R(r) Θ(θ) Φm(φ) Orbital magnetic quantum no Extracted Text from Image Top Left: Lz|m> = mħ|m> LzΦ(φ) = mħΦ(φ) Lz = |L|cosθ cosθ = Lz/|L| = mħ/|L| Top Right: Diagram of a coordinate system with x, y, and z axes. Center Left: [Lx, Ly] = [Lx, Lz] = [Ly, Lz] = 0 [Lx, Ly] ≠ 0, [Ly, Lz] ≠ 0, [Lz, Lx] ≠ 0 L^2 and Lz Can have simultaneous eigen fn Bottom Left: Let |l, m> be the simultaneous eigen fn of L^2 and Lz with eigen values lħ^2 and mħ respectively. L^2|l, m> = lħ^2|l, m> Lz|l, m> = mħ|l, m> L+L- + L-L+ = 2(L^2 - Lz^2) Bottom Right: = + L- = Lx - iLy L+ = Lx + iLy = L+ L-|γ, m> = |γ, m-1>, <γ, m|L+ = <γ, m+1| <γ, m|L+L-|γ, m> = <γ, m+1|γ, m>, = 0 using these Rel => <γ, m|(L+L- + L-L+)|<γ, m> ≥ 0 => <γ, m|(L^2 - Lz^2)|<γ, m> ≥ 0 => <γ, m|(<γ(γ+1)ħ^2 - m^2ħ^2)|γ, m> ≥ 0 => (γ(γ+1) - m^2)ħ^2 <γ, m|γ, m> ≥ 0 => <γ(γ+1) - m^2 ≥ 0 => <γ ≥ m^2

Related Documents

Eigen Value and Function of L

Eigen Value and Function of L - Page 1

Eigen Value and Function of L - Page 2

Eigen Value and Function of L - Page 3

Eigen Value and Function of L - Page 4

Eigen Value and Function of L - Page 5

of 5

Recommended Documents

Degeneracy Discriminant Factor

Degeneracy Discriminant Factor

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Maxwell's Velocity Distribution

Maxwell's Velocity Distribution

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Answer Key # 1 The Gibbs Free Energy

Answer Key # 1 The Gibbs Free Energy

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Answer Key

General Physics: Answer Key # 23

General Physics: Answer Key # 23

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Answer Key

Rules for Multiplication

Rules for Multiplication

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Basis Physics Notes

Basis Physics Notes

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Physically meaning of scalar product

Physically meaning of scalar product

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Eigen Value

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Anti Hermitten Operator

Anti Hermitten Operator

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Projection Operator

Projection Operator

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Singular and Non Singular Operator

Singular and Non Singular Operator

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Unitary Operator

Unitary Operator

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

New Documents from this Course

Modulus Study Documents

Modulus Study Documents

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Synchronous Counter

Synchronous Counter

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Full-Subtractor Coursework Materials

Full-Subtractor Coursework Materials

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Multiplier - Academic Notes

Multiplier - Academic Notes

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Comparator - Educational Materials

Comparator - Educational Materials

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Ripple Carry Adder

Ripple Carry Adder

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

MUX (Multiplexer) - Study Materials

MUX (Multiplexer) - Study Materials

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Assignment 2: Q and A

Assignment 2: Q and A

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Answer Key

Sequential Circuit - Study Note

Sequential Circuit - Study Note

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Latch (Temporary Storage of Bits)

Latch (Temporary Storage of Bits)

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Set-Reset Flip-Flop

Set-Reset Flip-Flop

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note

Delay Flip Flop

Delay Flip Flop

Santa Fe College PHY 2048 | General Physics 1 with Calculus

Lecture Note