Showing posts with label Quantum mechanics. Show all posts

Thursday, April 24, 2014

Quantum Tunneling Through a Barrier

Monday, June 8, 2009

Electric dipole-dipole interaction

Electric dipole-dipole interaction between f electron atoms

Estimate of electric dipole moment:

According to Yen et al.[1], the electric dipole moment can be estimated as:

p~e<r>V_c/D,

where e is the elctronic charge, <r>~10^-10 m is the mean radius of the 4f electronic orbit, V_c~500 cm^-1 is the characteristic crystal field splitting, and D~60000 cm^-1 [2] is the energy separation between (4f)ⁿ and (4f)^n-1(5d)' configuration.

Then p~1.3e-31 C.m

Then the dipole-dipole interaction is

W=\frac{ p_1 p_2}{4\pi \epsilon_0 R^3}

$eq=W=\frac{ p_1 p_2}{4\pi \epsilon_0 R^3}$

For example, if R=0.42 nm, we get

W=1.2 e-5 eV, or 0.01 meV.

[1] W. M. Yen et al., Physical Review 140, 1188 (1965).
[2] G. H. Dieke, and H. M. Crosswhite, Applied Optics 2, 675 (1963).

Thursday, May 21, 2009

From oscillator strength to spontaneo...

From oscillator strength to spontaneous decay rate:

Approach I

Using
quantum mechanics, one can derive the relation between the oscillator
strength calculated from the absorption coefficient and the transition
matrix element |x|². And the spontaneous decay rate is also decided by the matrix element |x|², therefore, one can find the decay rate from the absorption coefficient.

f=\frac{2m\omega}{\hbar}|x|^2

$eq=<br />f=\frac{2m\omega}{\hbar}|x|^2$

while spontaneous decay rate:
A=\frac{e\omega^3}{3\pi\hbar c^3\epsilon_0 n^2}|x|^2

$eq=A=\frac{e\omega^3}{3\pi\hbar c^3\epsilon_0 n^2}|x|^2$

Therefore,

\frac{A}{f}=\frac{\omega^2e^2}{6m\pi c^3\epsilon_0n^2}

$eq=\frac{A}{f}=\frac{\omega^2e^2}{6m\pi c^3\epsilon_0n^2}$

Approach II: Judd-Ofelt theory,

f=\frac{2m\omega}{\hbar}S

$eq=f=\frac{2m\omega}{\hbar}S$

A=\frac{\omega^3e^2n^2}{\hbar c^3\pi\epsilon_0}S

$eq=A=\frac{\omega^3e^2n^2}{\hbar c^3\pi\epsilon_0}S$

Therefore,

\frac{A}{f}=\frac{\omega^2e^2n^2}{2mc^3\pi\epsilon_0}

$eq=\frac{A}{f}=\frac{\omega^2e^2n^2}{2mc^3\pi\epsilon_0}$

One can see that, in two approaches, there is an factor of 3n^4.

I am not so sure why two are not the same, but at this moment I tend to believe the second approach because it is specialized just for this case.

Usage

In a real experiment, oscillator strength can be found using partial sum rule:
The dimensionless oscillator strength is:
f=\frac{2c}{N_e\hbar \pi \omega_p^2}\int_{E_2}^{E_1}n\alpha dE

$eq=f=\frac{2c}{N_e\hbar \pi \omega_p^2}\int_{E_2}^{E_1}n\alpha dE$

Here
\omega_p^2=\frac{e^2 \rho}{m\epsilon_0}

$eq=\omega_p^2=\frac{e^2 \rho}{m\epsilon_0}$

[Reference:]

Kumar G.A. et al, Journal of Luminescence vol. 99, p. 141-148 (2002)

Conservation of

Conservation of Σλ_i

Description:

For a Helbert space (wave function subspace) {φ_i} defined by the operator O whose eigenvalues are λ_i, corresponding to wavefunction φ_i.

Simply from linear algebra, Σλ_i is actually the trace of the operator matrix O, which is of course conserved, no matter what unitary transformation of the {φ_i} is taken to form new set of basis, say {ψ_i}.

To generalize this, Σλ_iⁿ, is also conserved, because Oⁿ also defines the space.

Example:

For a 4f electron multiplet, say, Nd³⁺, ⁴I_9/2, ΣJ_z,iⁿ is conserved and of course Σμ_z,iⁿ is conserved too because they differ only by a factor of gμ_B

Just to give a few numbers,

sqrt(3Σμ_z,i²/5)=

3.6 μ_B(theoretical value)
3.4, 3.5 and 3.8 from measurement on crystal field split ⁴I_9/2multiplet.

Saturday, July 26, 2008

Summary of 3d atomic orbitals

	Symmetry	Orbitals
Free atom		E₁ m=2, Ψ_3,2,2 = (1 / 81√(2π)) (Z/a)^7/2 r² e^-Zr/3a sin²θ e^i2φ m=1, Ψ_3,2,1 = (√2 / 81√π) (Z/a)^7/2 r² e^-Zr/3a sinθ cosθ e^iφ m=0, Ψ_3,2,0 = (1 / 81√(6π)) (Z/a)^7/2 r² e^-Zr/3a (3cos²θ -1) m=-1, Ψ_3,2,-1 = (√2 / 81√π) (Z/a)^7/2 r² e^-Zr/3a sinθ cosθ e^-iφ m=-2, Ψ_3,2,-2 = (1 / 81√(2π)) (Z/a)^7/2r² e^-Zr/3a sin²θ e^-i2φ
Octahedral / Tetrahedral	O_h / T_d	E₁ (e_g) x²-y²=(3,2,2)+(3,2,-2) sin²θ cos2φ z²=(3,2,0) (3cos²θ-1) E₂ (t_2g) xy=(3,2,2)-(3,2,-2) sin²θ sin2φ yz=(3,2,1)-(3,2,-1) sinθ cosθ sinφ xz=(3,2,1)+(3,2,-1) sinθ cosθ cosφ
Triangular bipyramid symmetry		E₁ x²-y², xy E₂ xz, yz E₃ z²

Angular momentum operator:
L_z=(ħ/i)∂/∂φ

Monday, June 30, 2008

Confusing matrix elements

When we discuss about the matrix element of optical transition, we often can use either of the following ways:

1) H₁=exE, where e is the electronic charge, E is the electric field
2) H1=epA/m, where p is the momentum of the electron and A is the vector potential.

Following approach 1):

transition rate:
W=|<φ_i|exE₀|φ_f>|² δ(E_i-E_f-ħω)=e²E₀²|<φ_i|x|φ_f>|²δ(E_i-E_f-ħω)

Power:
P=ħωe²E₀²|<φ_i|x|φ_f>|²δ(E_i-E_f-ħω)

Dielectric constant (imaginary part):
ε₂=P/(ε₀ωE₀²)=ħe²/ε₀|<φ_i|x|φ_f>|²δ(E_i-E_f-ħω)

However, if we use the approach 2)

transition rate:
W=|<φ_i|epA/m|φ_f>|² δ(E_i-E_f-ħω)=(e/m)²A₀²|<φ_i|p|φ_f>|²δ(E_i-E_f-ħω)

Power:
P=ħω(e/m)²A₀²|<φ_i|p|φ_f>|²δ(E_i-E_f-ħω)

Dielectric constant (imaginary part):
using the relation that E₀=iωA₀
ε₂=P/(ε₀ωE₀²)=ħ(e/m)²/(ε₀ω²)|<φ_i|p|φ_f>|²δ(E_i-E_f-ħω)

We can easily see that the function ε₂(ω) in the two cases are very different, which one is correct and why?

In all the books I read, approach 2) seems to be used, why?

Quantization of electromagnetic field:

Vector potential:

A_n(r)=sqrt(ħ/2ε₀ω)(a_n+a_n^†)u_n(r)