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Chapter 39

What is Physics?One of the long-standing goals of physics has been to understand the nature of atom. The development of quantum mechanics provided a framework for understanding this and many other mysteries. The basic premise of quantum mechanics is that moving particles (electrons, protons, etc) are best viewed as matter waves whose motions are governed by Schrödinger’s equation. Although this premise is also correct for massive objects (baseballs, cars. Planets, etc.) where classical Newtonian mechanics still predicts behavior correctly, it is more convenient to use classical mechanics in that regime. However, when particle masses are small, quantum mechanics provides the only framework for describing their motion.

Before applying quantum mechanics to the atomic structure, we will first explore some simpler situations. Some of these oversimplified examples, which previously were only seen in introductory textbooks, are now realized in real devices developed by the rapidly growing field of nanotechnology.

More About Matter Waves

39-

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In Ch. 16 we saw that two kinds of waves can be set up on a stretched string: traveling waves and standing waves.

• infinitely long string → traveling waves → frequency or wavelength can have any value.

•finite length string (e.g., clamped both ends) → standing waves → only discrete frequencies or wavelengths

→ confining a wave in a finite region leads to the quantization of its motion with discrete states each defined by a quantized frequency.

This observation also applies to matter waves.

• electron moving +x-direction and subject to no force (free particle) →wavelength (λ=h/p), frequency (f=v/λ), and energy (E=p2/2m) can have any reasonable value• atomic electron (e.g., valence): Coulomb attraction to nucleus →spatial confinement → electron can exist only in discrete states, each with a discrete energy

String Waves and Matter Waves

39-Confinement of wave leads to quantization: existence of discretestates with discrete energies. Wave can only have those energies.

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Fig. 16-23

One-Dimensional Trap:

Energies of a Trapped Electron

39-

, for 1, 2,32

nL nλ= = K

( ) sin , for 1, 2,32n

ny x A x nλ = =

K

n is a quantum number, identifying each state (mode)

Fig. 39-1

Fig. 39-2

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Finding the Quantized Energies

39-

Fig. 39-2

2 = 2 for 0 ,

2

p mK mE x Lh hp mE

λ

= < <

= =

Infinitely deep potential well

Fig. 39-3

22

2 , for 1, 2,38nhE n nmL

= =

K

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Energy Changes

39-

Fig. 39-4

high lowE E E∆ = −

high lowhf E E E= ∆ = −

Confined electron can absorb photon only if photon energy hf=∆E, the energy difference between initial energy level and a higher final energylevel.

Confined electron can emit photon only if photon energy hf=∆E, the energy difference between initial energy level and a lower final energy level.

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Probability of detection:

Wave Functions of a Trapped Electron

39-

( ) sin , for 1, 2,3nnx A x nLπψ = =

K

Fig. 39-6

( )( ) ( )

2probability probability densityof detection in width width

at position centered on position

n

p xx

dx dxx

x

ψ

=

( ) ( )2np x x dxψ=

( )2 2 2sin , for 1, 2,3nnx A x nLπψ = =

K

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To find probability that electron can be detected in any finite section of the well, e.g., between point x1 and x2, we must integrate p(x) between those points.

Wave Functions of a Trapped Electron, cont’d

39-

( )21

2

1

1 2

2 2

probability of detectionbetween and

sin

x

x

x

x

p xx x

nA x dxLπ

=

=

∫

∫

At large enough quantum numbers (n), the predictions of quantum mechanics merge smoothly with those of classical physics

Correspondence Principle:

Normalization:

( )21

2 1 (normalization equation)x

nxx dxψ =∫

The probability of finding the electron somewhere (if we search entire x-axis) is 1!

2A L→ =

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Zero-Point Energy:, In a quantum well, the lowest quantum number is 1 (n=0 means there is no electron in well), so the lowest energy (ground state) is E1which is also non-zero.

→confined particles must always have at least a certain minimum non-zero energy!

→since the potential energy inside the well is zero, the zero-point energy must come from the kinetic energy.

→a confined particle is never at rest!

Wave Functions of a Trapped Electron, cont’d

39-Fig. 39-3

Zero-Point Energy

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An Electron in a Finite Well

39-

Fig. 39-7

( )2 2

2 2

8+ - =0d m E U xdx hψ π ψ

Fig. 39-8

( ) 0x Lψ ≥ ≠

Fig. 39-9

Leakage into barriers →longer wavelengths →lower energies than infinite well

well

barrierbarrier

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Nanocrystallites: small (L~1nm) granule of a crystal trapping electron(s)

More Electron Traps

39-

Fig. 39-11

( )2 2 28E h mL n= Only photons with energy above minimum threshold energy Et (wavelength below a maximum threshold wavelength λt) can be absorbed by an electron in nanocrystallite. Since Et α 1/L2, the threshold energy can be increased by decreasing the size of the nanocrystallite.

tt t

c chf E

λ = =

Quantum Dots: electrons sandwiched in semiconductor layer→artificial atom with controllable number of electrons trapped→new electronics, new computing capabilities, new data storage capacity…

Quantum Corral: electrons “fenced in” by a corral of surrounding atoms.

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Rectangular Corral: infinite potential wells in the x and y directions

Two- and Three-Dimensional Electron Traps

39-

Fig. 39-13

222 2 22 2

, 2 2 2 28 8 8yx

nx ny x yx y x y

nnh h hE n nmL mL m L L

= + = +

Unlike a 1D well, in 2D certain energies may not be uniquely associated to a single state (nx, ny) since different combinations of nx,and ny can produce the same energy. Different states with the same energy are called degenerate. Fig. 39-15

If Lx=Ly

1239-

Fig. 39-14

Two- and Three-Dimensional Electron Traps

222 2

, , 2 2 28yx z

nx ny nzx y z

nnh nEm L L L

= + +

As in 2D, certain energies may not be uniquely associated to a single state (nx, ny, nz) since different combinations of nx,, ny,, and nz can produce the same (degenerate) energy.

Rectangular Box: infinite potential wells in the x, y, and z directions

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Hydrogen (H) is the simplest “natural” atom. Contain +e charge at center surrounded by –e charge (electron). Why doesn’t the electrical attraction between the two charges cause them to collapse together?

The Bohr Model of the Hydrogen Atom

39-Fig. 39-16

Balmer’s empirical (based only on observation) formula on absorption/emission of visible light for H

2 2

1 1 1 , for 3, 4,5, and 62

R nnλ

= − =

Bohr’s assumptions to explain Balmer formula

1) Electron orbits nucleus

2) The magnitude of the electron’s angular momentum L is quantized

, for 1, 2,3,L n n= =h K

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Coulomb force attracting electron toward nucleus

Orbital Radius is Quantized in the Bohr Model

39-

1 22

q qF k

r=

2 2

20

14

e vF ma mr rπε

= − = = −

Quantize angular momentum l : sin nrmv rmv n vrm

φ= = = → = hl h

Substitute v into force equation:2

202 , for 1, 2,3,

hr n nmeε

π= = K 2 , for 1, 2,3,r an n= = K

Where the smallest possible orbital radius (n=1) is called the Bohr radius a:2

1002 5.291772 10 m 52.92 pm

hameε

π−= = × ≈

Orbital radius r is quantized and r=0 is not allowed (H cannot collapse).

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The total mechanical energy of the electron in H is:

Orbital Energy is Quantized

39-

221

2 20

14

eE K U mvrπε

= + = + −

Solving the F=ma equation for mv2 and substituting into the energy equation above:

The energy of the electron (or the entire atom if nucleus at rest) in a hydrogen is quantized with allowed values En.

2

0

18

eErπε

= −

Substituting the quantized form for r:4

2 2 20

1 for 1, 2,3,8nmeE n

h nε= − = K

18

2 2

2.180 10 J 13.60 eV= , for 1, 2,3,nE nn n

−×= − = K

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The energy of a hydrogen atom (equivalently its electron) changes when the atom emits or absorbs light:

Energy Changes

39-

Substituting f=c/λ and using the energies En allowed for H:

This is precisely the formula Balmer used to model experimental emission and absorption measurements in hydrogen! However, the premise that the electron orbits the nucleus is incorrect! Must treat electron as matter wave.

2 2low high

1 1 1Rn nλ

= −

high lowhf E E E= ∆ = −

4

2 3 2 20 high low

1 1 18

meh c n nλ ε

= − −

Where the Rydberg constant4

7 -12 30

1.097373 10 m8

meRh cε

= = ×

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The potential well that traps an electron in a hydrogen atom is:

Schrödinger’s Equation and the Hydrogen Atom

39-

Fig. 39-17

( )2

04eU r

rπε−

=Energy Levels and Spectra of the Hydrogen Atom:

Fig. 39-18

Can plug U(r) into Schrödinger’s equation to solve for En.

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Principal quantum number n → energy of state

Orbital quantum number l → angular momentum of state

Orbital magnetic quantum number ml → orientation of angular momentum of state

Quantum Numbers for the Hydrogen Atom

39-

Quantum Numbers for the Hydrogen Atom

Symbol Name Allowed Values

n Principal quantum number 1, 2, 3, …

l Orbital quantum number 0, 1, 2, …, n-1

ml Orbital magnetic quantum number -l, -(l-1), …+(l-1), +l

Table 39-2

For ground state, since n=1→ l=0 and ml =0

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Solving the three-dimensional Schrödinger equation and normalizing the result:

Wave Function of the Hydrogen Atom’s Ground State

39-

( ) 32

1= (ground state)rar ea

ψπ

−

( ) ( )2probability of detection volume probability

in volume density volume centered at radius r at radius r

ndV r dVψ =

( )24dV r drπ=

( )2 2 23probability of detection

4 in volume centered at radius r

r andV r dV e r dra

ψ − = =

20

Radial probability density P(r):

Wave Function of the Hydrogen Atom’s Ground State, cont’d

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( ) ( ) ( )2radial probability volume probability

radial density density volume

width at radius at radius r

nP r r dVdrr

ψ

=

( ) ( )2P r dr r dVψ=

( ) 2 234 (radial probability density,

hydrogen atom ground state)

r aP r r ea

−=

( )0

1P r dr∞

=∫The probability of finding the electron somewhere (if we search all space) is 1!

21Fig. 39-21

Wave Function of the Hydrogen Atom’s Ground State, cont’d

39-Fig. 39-20

( )2 rψ

Probability of finding electron within a small volume at a given position

Probability of finding electron within a within a small distance from a given radius

( )P r

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Solving the three-dimensional Schrödinger equation and

Hydrogen Atom States with n=2

39-

Quantum Numbers for Hydrogen Atom States with n=2

n l ml

2 0 02 1 +12 1 02 1 -1

Table 39-3

23Fig. 39-23

Hydrogen Atom States with n=2, cont’d

39-

Fig. 39-22

( )2 for 2, 0, and 0

r nm

ψ = ==l

l

Fig. 39-24

( )2 2, 1r nψ = =l ( )2 2, 1r nψ = =l

Direction of z-axis completely arbitrary

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As the principal quantum number increases, electronic states appear more like classical orbits.

Hydrogen Atom States with n>>1

39-

Fig. 39-25

( ) for 45, 1 44P r n n= = − =l