Chapter 39 More About Matter Waves

What Is Physics?One of the long-standing goals of physics has been to understand the nature of the 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.

(39-1)

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 (=h/p), frequency (f=v/), 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.

39-2 String Waves and Matter Waves

Confinement of wave leads to quantization: existence of discrete states with discrete energies. Wave can only have those energies. (39-2)

Fig. 16-23

One-Dimensional Trap:

39-3 Energies of a Trapped Electron

, for 1,2,32

nL n

sin , for 1,2,3n

ny x A x n

L

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

Fig. 39-1

Fig. 39-2

(39-3)

Finding the Quantized Energies

Fig. 39-2

2 = 2 for 0 ,

2

p mK mE x L

h h

p mE

Infinitely deep potential well (infinte potential well)

Fig. 39-3

22

2 , for 1,2,3

8n

hE n n

mL

(39-4)

Energy Changes

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 energy level.

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

(39-5)

Probability of detection:

39-4 Wave Functions of a Trapped Electron

sin , for 1,2,3n

nx A x n

L

Fig. 39-6

2probability

probability densityof detection in width width

at position centered on position

n

p xx

dx dxx

x

2np x x dx

2 2 2sin , for 1,2,3n

nx A x n

L

(39-6)

To find the probability that an 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

2

1

2

1

1 2

2 2

probability of detection

between and

sin

x

x

x

x

p xx x

nA x dx

L

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

Correspondence Principle:

Normalization:

2

1

2 1 (normalization equation)x

nxx dx

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

2A L

(39-7)

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 E1, which is also nonzero.

→confined particles must always have at least a certain minimum nonzero 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

Fig. 39-3

Zero-Point Energy

(39-8)

39-5 An Electron in a Finite WellFig. 39-7

2 2

2 2

8+ - =0

d mE U x

dx h

Fig. 39-8

0x L

Fig. 39-9

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

well

barrierbarrier

(39-9)

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

39-6 More Electron Traps

Fig. 39-11

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

tt t

c ch

f E

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.

(39-10)

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

39-7 Two- and Three-Dimensional Electron Traps

Fig. 39-13

222 2 22 2

, 2 2 2 28 8 8yx

nx ny x yx y x y

nnh h hE n n

mL mL m L L

Unlike a 1D well, in 2D certain energies may not be uniquely associated with 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

(39-11)

Fig. 39-14

Two- and Three-Dimensional Electron Traps

222 2

, , 2 2 28yx z

nx ny nzx y z

nnh nE

m L L L

As in 2D, certain energies may not be uniquely associated with 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

(39-12)

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

39-8 The Bohr Model of the Hydrogen Atom

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 6

2R n

n

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 (39-13)

Coulomb force attracting electron toward nucleus

Orbital Radius Is Quantized in the Bohr Model

1 22

q qF k

r

2 2

20

1

4

e vF ma m

r r

Quantize angular momentum l : sinn

rmv rmv n vrm

Substitute v into force equation:2

202

, for 1, 2,3,h

r n nme

2 , for 1, 2,3,r an n

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

1002

5.291772 10 m 52.92 pmh

ame

Orbital radius r is quantized and r = 0 is not allowed (H cannot collapse).(39-14)

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

The total mechanical energy of the electron in H is:

Orbital Energy Is Quantized

221

2 20

1

4

eE K U mv

r

Solving the F = ma equation for mv2 and substituting into the energy equation above: 2

0

1

8

eE

r

Substituting the quantized form for r:4

2 2 20

1 for 1, 2,3,

8n

meE n

h n

18

2 2

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

n n

(39-15)

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.

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

Energy Changes

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

2 2low high

1 1 1R

n n

high lowhf E E E

4

2 3 2 20 high low

1 1 1

8

me

h c n n

where the Rydberg constant4

7 -12 30

1.097373 10 m8

meR

h c

(39-16)

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

39-9 Schrödinger’s Equation and the Hydrogen Atom

Fig. 39-17

2

04

eU r

r

Energy Levels and Spectra of the Hydrogen Atom:

Fig. 39-18

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

(39-17)

Principal quantum number n → energy of state

Orbital quantum number l → angular momentum of state

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

Quantum Numbers for the Hydrogen Atom

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

(39-18)

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

Wave Function of the Hydrogen Atom’s Ground State

32

1= (ground state)rar e

a

2

probability of detection volume probability

in volume density volume

centered at radius at radius ndV r dV

r r

24dV r dr

2 2 23

probability of detection4

in volume

centered at radius

r andV r dV e r dr

ar

(39-19)

Radial probability density P(r):

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

2

radial probability volume probabilityradial

density density volume width

at radius at radius nP r r dV

drr r

2P r dr r dV

2 23

4 (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!

(39-20)

Fig. 39-21

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

Fig. 39-20

2 r

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

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

P r

(39-21)

Solving the three-dimensional Schrödinger equation.

Hydrogen Atom States with n = 2

Quantum Numbers for Hydrogen Atom States with n = 2

n l ml

2 0 02 1 +12 1 02 1 -1

Table 39-3

(39-22)

Fig. 39-23

Hydrogen Atom States with n = 2, cont’d

Fig. 39-22

2 for 2, 0,

and 0

r n

m

Fig. 39-24

2 2, 1r n 2 2, 1r n

Direction of z-axis completely arbitrary

(39-23)

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

Hydrogen Atom States with n >> 1

Fig. 39-25

for 45, 1 44P r n n

(39-24)