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The Bohr atom and the Uncertainty Principle

Previous Lecture:

Matter waves and De Broglie wavelength

The Bohr atom

This Lecture:

More on the Bohr Atom

The H atom emission and absorption spectra

Uncertainty Principle

Wave functions

MTE 3 Contents, Apr 23, 2008

! Ampere’s Law and force between wires with current (32.6, 32.8)

! Faraday’s Law and Induction (ch 33, no inductance 33.8-10)

! Maxwell equations, EM waves and Polarization (ch 34, no 34.2)

! Photoelectric effect (38.1-2-3)

! Matter waves and De Broglie wavelength (38.4)

! Atom (37.6, 37.8-9, 38.5-7)

! Wave function and Uncertainty (39)

2

Requests for alternate exams should be sent before Apr. 20

HONOR LECTURE

! PROF. RZCHOWSKI

! COLOR VISION

3

Swinging a pendulum: the classical picture

Larger amplitude, larger energy

Small energy Large energy

d

! Potential Energy E=mgd

! E.g. (1 kg)(9.8 m/s2)(0.2 m) ~ 2 Joules

d

The quantum mechanics scenario

! Energy quantization: energy can have only certain discrete values

Energy states are separated by !E = hf.

f = frequencyh = Planck’s constant= 6.626 x 10-34 Js

d

Suppose the pendulum has

Period = 2 secFrequency f = 0.5 cycles/sec

!Emin=hf=3.3x10-34 J << 2 J

Quantization not noticeable

at macroscopic scales6

Bohr Hydrogen atom

! Hydrogen-like atom has one electron and Z protons

! the electron must be in an allowed orbit.

! Each orbit is labeled by the quantum number n. Orbit radius = n2ao/Z and

a0 = Bohr radius = 0.53 !

! The angular momentum of each orbit is quantized Ln = n ℏ

! The energy of electrons in each orbit is En = -13.6 Z2 eV/n2

Radius and Energy levels of H-like ions

7

!

F = kZe

2

r2

= mv2

r

mvr = nh

Total energy:

!

E =p2

2m"kZe

2

r# E = "

1

2

k2Z2me

4

n2h2

= "13.6eVZ2

n2

!

r = n2 h

2

mkZe2

"

# $

%

& ' =

n2

Za0

Charge in nucleus Ze (Z = number of protons)

Coulomb force centripetal

The minimum energy (binding energy) we need to provide to ionize an H atom in its ground state (n=1) is 13.6 eV (it is lower if the atom is in some excited state and the electron is less bound to the nucleus).

“Correspondence principle”: quantum mechanics must agree with classical results when appropriate (large quantum numbers)

!

r = n2a0"#

Put numbers in

8

!

F = kZe

2

r2

= mv2

r

mvr = nh

!

kZe

2

r= mv

2

Total energy = kinetic energy + potential energy = kinetic energy + eV

!

r = kZe

2

mv2

!

v =nh

mr

Substitute (2) in (1):

(1)

(2)

!

r = kZe

2m2r2

mn2h2" r =

n2

Z

h2

me2k

=n2

Z

(1.055 #10$34)2

9.11#10$31# (1.6 #10

$19)2# 9 #10

9=n2

Z0.53#10

$10m

!

r =n2

Za0

!

E =p2

2m" k

Ze2

r=mv

2

2" k

Ze2

r= "k

Ze2

2r= "k

Ze2

2#Zme

2k

n2h2

= "Z2

n2

k2me

4

2h2

= "13.6eVZ2

n2

!

E = "Z2

n2

(9 #109)2# 9.11#10

"31# (1.6 #10

"19)4

2 # (1.055 #10"34)2

= "Z2

n22.2 #10

"18J = "

Z2

n213.6eV

a0 = Bohr radius = 0.53 x 10-10 m = 0.53 !

Z=n. of protons in

nucleus of H-like

atom

Photon emission question

• An electron can jump between the energy levels of on H atom. The 3 lower levels are part of the sequence En = -13.6/n2 eV with n=1,2,3. Which of the following photons could be emitted by the H atom?

• A. 10.2 eV

• B. 3.4 eV

• C 1.0 eV

9E1=-13.6 eV

E2=-3.4eV eV

E3=-1.5 eV

!

13.6 " 3.4 =10.2eV

13.6 "1.5 =12.1eV

3.4 "1.5 =1.9eV10

Energy conservation for Bohr atom

! Each orbit has a specific energy E

n=-13.6/n2

! Photon emitted when electron jumps from high energy Eh to low energy orbit El (atom looses energy).

El – E

h = h f

! Photon absorption induces electron jump from low to high energy orbit (atom gains energy).

El – E

h = h f

Bohr successfully explained H-like atom absorption and emission spectra but not heavier atoms

<0

>0

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Example: the Balmer series

! All transitions terminate at the n=2 level

! Each energy level has energy En=-13.6 / n2 eV

! E.g. n=3 to n=2 transition! Emitted photon has energy

! Emitted wavelength

!

Ephoton = "13.6

32

#

$ %

&

' ( " "

13.6

22

#

$ %

&

' (

#

$ %

&

' ( =1.89 eV

!

Ephoton = hf =hc

", " =

hc

Ephoton

=1240 eV # nm

1.89 eV= 656 nm

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Hydrogen spectra

• Lyman Series of emission lines:

• Balmer:

Hydrogen

n=2,3,4,..

Use E=hc/"

R = 1.096776 x 107 /m

For heavy atoms R! = 1.097373 x 107 /m

Rydberg-Ritz

!

1

"= R

1

22#1

n2

$

% &

'

( ) n=3,4,..

Suppose an electron is a wave…

! Here is a wave:

…where is the electron?

! Wave extends infinitely far in +x and -x direction

"

x

!

" =h

p

Analogy with sound

! Sound wave also has the same characteristics

! But we can often locate sound waves

! E.g. echoes bounce from walls. Can make a sound pulse

! Example:

! Hand clap: duration ~ 0.01 seconds

! Speed of sound = 340 m/s

! Spatial extent of sound pulse = 3.4 meters.

! 3.4 meter long hand clap travels past you at 340 m/s

Beat frequency: spatial localization

! What does a sound ‘particle’ look like?

! One example is a ‘beat frequency’ between two notes

! Two sound waves of almost same wavelength added.

Constructive interference

Large amplitude

Constructive interference

Large amplitude

Destructive interference

Small amplitude

Creating a wave packet out of many waves

f1 = 440 Hz

f2= 439 Hz

f1+f2

440 Hz + 439 Hz + 438 Hz

440 Hz + 439 Hz + 438 Hz + 437 Hz + 436 Hz

Soft Loud Soft Loud

Beat

! Construct a localized particle by adding together waves with slightly different wavelengths (or frequencies).

Tbeat = !t = 1/!f =1/(f2-f1)

A non repeating wave...like a particle

! Six sound waves with slightly different frequencies added together

-8

-4

0

4

8

-15 -10 -5 0 5 10 15J

!t

•We do not know the exact frequency:

– Sound pulse is comprised of several frequencies in the range !f, we are uncertain about the exact frequency

–The smallest our uncertainty on the frequency the larger !t, that is how long the wave packet lasts

–

!

"f #"t $1

During !t the wave packet length is

this the uncertainty on the particle location

Heisenberg’s Uncertainty principle

!

"f #"t $1

x-8

-4

0

4

8

-15 -10 -5 0 5 10 15J

!x

!

"x = v"t =px

m"t

!

f =v

"=px

m#px

h=px2

mh$%f =

2px

hm%px

!

"f #"t =2px

hm"px #

m

px"x =

2

h"px #"x $1

!

"px #"x $h

2

! The packet is made of many frequencies and wavelengths

! The uncertainty on the wavelength is !"

! de Broglie: " = h /p => each of the components has slightly

different momentum.

! The uncertainty in the momentum is:

Uncertainty principle

19

The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. --Heisenberg, uncertainty paper, 1927

Particles are waves and we can only determine a probabilitythat the particle is in a certain region of space. We cannot say exactly where it is!

Similarly since E = hf

!

"px #"x $h

2

!

"E #"t $h

2

Uncertainty principle question

! Suppose an electron is inside a box 1 nm in width. There is some uncertainty in the momentum of the electron. We then squeeze the box to make it 0.5 nm. What happens to the momentum?

! A. Momentum becomes more uncertain

! B. Momentum become less uncertain

! C. Momentum uncertainty unchanged

20

!

I" A2

!

D = D1 + D2 = Asin(kr1"#t) + Asin(kr2 "#t)

Electron Interference

! superposition of 2 waves with same frequency and amplitude

! Intensity on screen and probability of detecting electron are connected

Computer

simulation

photograph

x

!

I(x) = Ccos2 "dx

#L

$

% & '

( )

!

P(x)dx =N(in dx at x)

Ntot

"Energy(in dx at x) / t

Ntothf / t=

=I(x)Hdx

Ntothf" A(x)

2dx

Wave function

! When doing a light interference experiment, the probability that photons fall in one of the strips around x of width dx is

! The probability of detecting a photon at a particular point is directly proportional to the square of light-wave amplitude function at that point

! P(x) is called probability density (measured in m-1)

! P(x)∝|A(x)|2 A(x)=amplitude function of EM wave

! Similarly for an electron we can describe it with a wave function #(x) and P(x)∝| #(x)|2 is the probability

density of finding the electron at x

22 x

H

23

Wave Function of a free particle

• #(x) must be defined at all points in space and be single-valued

• #(x) must be normalized since the particle must be somewhere in the entire space

• The probability to find the particle between xmin and xmax is:

• #(x) must be continuous in space– There must be no discontinuous jumps in the value of the

wave function at any point

!

P(xmin " x " xmax ) = #(x)2dx

xmin

xmax

$

Where is most propably the electron?

24

your HW

Suppose now that a is another number, a = 1/2, what is the probability that the electron is between 0 and 2?

A. 1/2B. 1/4 C. 2

How do I get a? from normalization condition

area of triangle =(a x 4)/2 = 1

Probability = area of tringle between 0,2 = (a x 2)/2 =

1/2

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Wave Function of a free particle• #(x) may be a complex function or a real function, depending on the

system

• For example for a free particle moving along the x-axis #(x) = Aeikx

– k = 2!/" is the angular wave number of the wave representing the particle

– A is the constant amplitude

Remember: complex number

imaginary unit