Lecture 9Lecture 9
Atoms, atomic transitions, & neon lighting
Absorption in Solar SpectrumAtomic transitions
Neon LightingLasers
Recap from last week: Electron Wavefunctions
iEtSteady-state total wavefunction:
iEtxtx, )exp()(
E energy of the electron E=energy of the electron
t=time
ψ(x) = electron wavefunction that describes only the spatially ψ(x) electron wavefunction that describes only the spatially behavior
Experimentally, we measure the probability of finding an electron in a given position at time t (like an intensity):
22 |)(||)(| zy,x,tz,y,x,
S h di ’ i f di i
Time independent Schrodinger equation
022
VEmd
Schrodinger’s equation for one dimension
022 VEdx
Schrondinger’s equation for three dimensions
0)(222
2
2
2
2
2
VEm
zyx zyx
A mathematical “crank”: we input the potential V of the electron (i.e., the ‘force’ it experiences, F=-dV/dx), and can obtain the
electron energies E and their wavefunctions / probability
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)
electron energies E and their wavefunctions / probability distributions.
Example 1: electrons in a 1D box
axnAxn sin)(
2222
Wavefunction:
2
22
2
22
82)(
manh
manEn
Electron energy in an infinite PE well:
2 )12( nh
Energy separation in an infinite PE well:
21 8)12(
manhEEE nn
Example 1: electrons in a 1D box
Wavefunction: Probability:
Atoms: A 3D Quantum well
Electron confined in three dimensions by a three-dimensional infinite PE box. yEverywhere inside the box, V = 0, but outside, V = . The electron cannot escape
from the box.
Quantum mechanics in 3D: 3 quantum numbers
Electron wavefunction in infinite PE well
znynxn
czn
byn
axnAz y, x,nnn
321 sinsinsin)(321
Electron energy in infinite PE box
2
22
2
23
22
21
2
88321
NhnnnhE nnn
22 88321 mamannn
23
22
21
2 nnnN 321 nnnN
Atoms: electrons in a 3D spherical box
2m 022
2 VEm
The electron in the hydrogenic atom is atom is attracted by a central force that is always directed toward the positive nucleusis always directed toward the positive nucleus.
Spherical coordinates centered at the nucleus are used to describe the positionof the electron. The PE of the electron depends only on r.
Atoms: electrons in a 3D spherical box
The electron’s potential energy V(r) in a hydrogenic atom (i.e., just one electron) is used in the Schrödinger equation.just one electron) is used in the Schrödinger equation.
Recall: r2=x2+y2+z2
2m
Solutions to Schrodinger’s equation in an atom
022
2 VEm
rZerV4
)(2
ro4
:2 f
)()()( ),()(),,( YrRr
R: Radial wavefunction – depends on two quantum numbers, “n” and “l”d w d p d w q b , dY: Angular wavefunction – depends on another quantum number, “ml”
(A fourth quantum number, also in Y, arises from relativity: “ms”)
Radial wavefunctions, R(r)
Radial wavefunctions of the electron in a hydrogenic atom for various n and values.
1 2 3 (j t lik ti l i b )n=1,2,3,…(just like particle in a box)=0,1,2,3,…(n-1) =0: “s” =1: “p” =2: “d” =3: “f”
Radial Probability
r2 |Rn,2| gives the radial probability density., i.e., where are we most
likely to find the electron around the nucleuslikely to find the electron around the nucleus
Radial ProbabilityBohr radius (a0): 0.0529nmBohr radius (a0): 0.0529nm
r2 |Rn,2| gives the radial probability density., i.e., where are we most
likely to find the electron around the nucleuslikely to find the electron around the nucleus
Angular Wavefunctions, Y(θ,φ)
The polar plots of Y(, ) for 1s and 2p states.
Angular Probability Distribution
The angular dependence of the probability distribution, which is proportional to | Y(, )|2.
Total Wavefunction: ψ(r, θ, φ)=R(r)Y(θ,φ)
n=1,2,3,…(just like particle in a box)(j p )=0,1,2,3,…(n-1) =0: “s” =1: “p” =2: “d” =3: “f”m =- …,0,…
Knowing ψ we can use the Schrodinger equation to find the electron
Electron energies
Knowing ψ, we can use the Schrodinger equation to find the electron energies.
Just like electrons in a 1D or 3D quantum well, the electron energy in the hydrogenic atom is quantized.
24Z222
24
8 nhZmeEn
8 nho
Ionization energy of hydrogen: energy required to remove the
(Z is atomic number, n is the quantum number, 1,2,3,…)
184me
Ionization energy of hydrogen: energy required to remove the electron from the ground state in the H-atom
eV13.6J1018.28
1822
hmeE
oI
Electron energies in 1-electron atoms
Energies are more closely spaced for
higher n
Radial electron ‘position’ & ionization energies in a hydrogenic atom (Z≥1)
an 2
y g ( )
Zanr o
max Z
2
2effective eV)6.13(ZE nI 2, nnI
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw‐Hill, 2005)
Example of Zeff: Li as a ‘hydrogenic atom’
The Li atom has a nucleus with charge +3e, 2 electrons in the K shell , which is closed, and one electron in the 2s orbital. (b) A simple view of (a) would be one ( ) p ( )
electron in the 2s orbital that sees a single positive charge, Z = 1
The simple view Z = 1 is not a satisfactory description for the outer electron because it has a probability distribution that penetrates the inner shell We can because it has a probability distribution that penetrates the inner shell. We can
instead use an effective Z, Zeffective = 1.26, to calculate the energy of the outer electron in the Li atom.
Energy transitions can occur via photons
Absorption of a photon
Energy transitions can occur via photons
Absorption of a photon
Emission of a photon
Example: Solar Spectrum
1829: Josef von Fraunhofer1829: Josef von Fraunhoferλdark
1=656.3 nmλdark
2=486.1 nm
eVEn
Enh
meZEo
n 6.13)1(8 121222
42
Convenient conversion: λ [eV]= 1241.341/λ [nm]
Example: Solar Spectrum
1829: Josef von Fraunhoferλdark
1=656.3 nm = 1.89eVdark(n=3 to n=2)
λd k2=486 1 nm = 2 55 eVλdark =486.1 nm = 2.55 eV
(n=4 to n=2)
eVEn
Enh
meZEo
n 6.13)1(8 121222
42
Energy transitions can also occur via collisions
An atom can become excited by a collision with another atom.An atom can become excited by a collision with another atom.
Energy transitions can also occur via collisions
An atom can become excited by a collision with another atomAn atom can become excited by a collision with another atom.When it returns to its ground energy state, the atom emits a photon.
Neon lighting occurs via quantum transitions!
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