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1
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
11
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
12
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
25
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
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