Space Quantization - Edubirdie

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

2024
Views:

20

Pages:

5
Author:

Jabin F.

Space Quantization ⇒ Position space quantization Angular momentum quantization l = 1 ⇒ m = +1, 0, -1 Lz / |L| = mħ / √l(l+1)ħ = ±1/√2, 0 cosθ = ±1/√2, 0 Sphere change in this form in 3D. θ = cos⁻¹(±1/√2), cos⁻¹(0) all portion in not allowed in Φ, m. (angle not fixed about x, y) Prob. Find the expectation value of the following operators in a state |j, m>, that is simultaneous eigenstate of L^2 and Lz with eigenvalues j(j+1)ħ^2 and mħ respectively. (i) Lz (ii) Lx (iii) Ly (iv) L+ (v) L- (vi) Lz^2 (vii) L+^2 (viii) L-^2 (ix) L^2 (x) LzL+ (xi) LzL- (xii) LxLz (xiii) LyLz (xiv) LxLy (xv) LyLx (xvi) LxLy + LyLx (i) = = mħ (ii) = <(L+ + L-)/2> = 0 (iii) = <(L+ - L-)/2i> = 0 (iv) = = sqrt[(j-m)(j+m+1)]ħ = 0 (v) = = sqrt[(j+m)(j-m+1)]ħ = 0 (vi) = = (mħ)^2 = m^2ħ^2 (vii) = = sqrt[(j-m)(j+m+1)]ħ = (sqrt[(j-m)(j+m+1)]ħ)^2 = (...) (...) delta_l,l delta_m,m+2 = 0 ⇒ = 0 (ix) = = = 1/4( + + ) = 1/4(0 + 0 + ) = 1/2( - ) = 1/2(j(j+1)ħ^2 - m^2ħ^2) = (l(l+1) - m^2)ħ^2 / 2 Similarly, = (l(l+1) - m^2)ħ^2 / 2 = = = ħ^2(j(j+1) - m(m-1)) = l(l+1) - m(m-1)ħ^2 = = l(l+1) - m(m+1)ħ^2 = 2(l(l+1) - m^2)ħ^2 = = mħ = mħ * 0 = 0 = 0 (xiv) = mħ = 0 (xv) = mħ = 0 (xvi) = mħ = 0 (xvii) = = 1/4i^2() = -1/4() = -1/4(- - ) = 1/4( + ) = 1/4 * 2(l(l+1) - m^2)ħ^2 = (l(l+1) - m^2)ħ^2 / 2 (xviii) = - = -(l(l+1) - m^2)ħ^2 / 2 (xix) = 0 |ψ'> = R|ψ> A' = RAR† R = Rotational operator that rotates by an angle φ about an axis n counterclockwise = e^(-iφn•J/ħ)

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