Download The Bohr atom and the Uncertainty Principle

The Bohr atom and the
Uncertainty Principle
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)
!
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
!
!
Requests for alternate exams should be sent
before Apr. 20
1
2
Swinging a pendulum: the classical picture
HONOR LECTURE
!
!
Larger amplitude, larger energy
PROF. RZCHOWSKI
COLOR VISION
Small energy
Large energy
d
d
!
!
Potential Energy E=mgd
E.g. (1 kg)(9.8 m/s2)(0.2 m) ~ 2 Joules
3
The quantum mechanics scenario
!
Energy quantization: energy can have only certain discrete
values
Energy states are separated by !E = hf.
f = frequency
h = Planck’s constant= 6.626 x 10-34 Js
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
!
The energy of electrons in each orbit
is En = -13.6 Z2 eV/n2
Suppose the pendulum has
Period = 2 sec
Frequency f = 0.5 cycles/sec
!Emin=hf=3.3x10-34 J << 2 J
d
Quantization not noticeable
at macroscopic scales
6
Radius and Energy levels of H-like ions
Put numbers in
Charge in nucleus Ze (Z = number of protons)
Coulomb force centripetal
2
Ze
v2
=
m
r2
r
mvr = nh
F=k
" h2 % n 2
r = n 2$
= a0
2'
# mkZe & Z
Ze 2
v2
F=k 2 =m
r
r
mvr = nh
!
2
Total energy: E =
!
2
2
2
4
2
p
kZe
1 k Z me
Z
"
# E ="
= "13.6eV 2
2m
r
2 n 2h2
n
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 = n 2 a0 " #
2
nh
(2)
mr
r=k
Ze 2
mv 2
Z=n. of protons in
nucleus of H-like
atom
!
2
Ze m r
n h
n2
(1.055 #10$34 ) 2
n2
"r=
=
= 0.53 #10$10 m
!
mn 2 h 2
Z me 2 k Z 9.11#10$31 # (1.6 #10$19 ) 2 # 9 #10 9 Z
n2
r = a0
a0 = Bohr radius = 0.53 x 10-10 m = 0.53 !
Z
r=k
!
!
Total energy = kinetic energy + potential energy = kinetic energy + eV
!
E=
p2
Ze 2 mv 2
Ze 2
Ze 2
Ze 2 Zme 2 k
Z 2 k 2 me 4
Z2
"k
=
"k
= "k
= "k
# 2 2 =" 2
= "13.6eV 2
2m
r
2
r
2r
2
n h
n 2h 2
n
E ="
7
Z 2 (9 #10 9 ) 2 # 9.11#10"31 # (1.6 #10"19 ) 4
Z2
Z2
= " 2 2.2 #10"18 J = " 2 13.6eV
8
n2
2 # (1.055 #10"34 ) 2
n
n
!
!
!
Energy conservation for Bohr 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
E3=-1.5 eV
• C 1.0 eV
13.6 " 3.4 = 10.2eV
13.6 "1.5 = 12.1eV
3.4 "1.5 = 1.9eV
2 2
v=
(1)
Ze 2
= mv 2
r
!
Substitute (2) in (1):
2
k
E2=-3.4eV eV
E1=-13.6 eV
9
!
Each orbit has a specific energy
En=-13.6/n2
!
Photon emitted when electron
jumps from high energy Eh to low
energy orbit El (atom looses energy).
El – Eh = h f
<0
!
Photon absorption induces electron
jump from
low to high energy orbit (atom gains energy).
El – Eh = h f
>0
Bohr successfully explained H-like atom absorption and
emission spectra but not heavier atoms
10
!
Hydrogen spectra
Example: the Balmer series
!
!
!
• Lyman Series of emission lines:
All transitions terminate at
the n=2 level
Each energy level has
energy En=-13.6 / n2 eV
• Balmer:
Emitted photon has energy
!
Use E=hc/"
## 13.6 & # 13.6 &&
E photon = %%" 2 ( " %" 2 (( = 1.89 eV
$$ 3 ' $ 2 ''
!
!
Emitted wavelength
E photon = hf =
Rydberg-Ritz
hc
hc
1240 eV # nm
, "=
=
= 656 nm
"
E photon
1.89 eV
11
!
n=3,4,..
R = 1.096776 x 107 /m
E.g. n=3 to n=2 transition
!
$1 1'
1
= R& 2 # 2 )
%2
"
n (
n=2,3,4,..
Hydrogen
For heavy atoms R! = 1.097373 x 107 /m
12
Suppose an electron is a wave…
Analogy with sound
!
!
!
Here is a wave:
"=
h
p
!
x
"
!
…where is the electron?
! Wave extends infinitely far in +x and -x direction
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
Creating a wave packet out of many waves
Beat frequency: spatial localization
Tbeat = !t = 1/!f =1/(f2-f1)
What does a sound ‘particle’ look like?
! One example is a ‘beat frequency’ between two notes
! Two sound waves of almost same wavelength added.
!
f1 = 440 Hz
f2= 439 Hz
Soft
Loud
Soft
Loud
f1+f2
Beat
440 Hz +
439 Hz +
438 Hz
Constructive
interference
Large
amplitude
440
439
438
437
436
Constructive
interference
Destructive
interference
Small
amplitude
Large
amplitude
Heisenberg’s Uncertainty principle
!
Six sound waves with slightly different frequencies added together
!
•We do not know the exact frequency:
8
!
– 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
–
8
4
"f # "t $ 1
0
-4
-8
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.
During !t the wave
packet length is
p
"x = v"t = x "t
m
this the uncertainty on
!x
x
the particle location
The uncertainty in the momentum is:
-15
-10
-5
0
5
10
15
J
4
!
0
"f # "t
!t
-15
-10
-5
0
v px px px2
2p
=
#
=
$ %f = !x %px
" m h mh
hm
2 px
m
2
"f # "t =
"px # "x = "px # "x $ 1
$1
hm
px
h
f =
!
-4
-8
5
10
+
+
+
+
Construct a localized particle by adding together waves with
slightly different wavelengths (or frequencies).
!
A non repeating wave...like a particle
!
Hz
Hz
Hz
Hz
Hz
!
15
"px # "x $
J
!
!
!
h
2
Uncertainty principle
Uncertainty principle question
The more precisely the position is
determined, the less precisely the
momentum is known in this instant,
and vice versa.
!
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
--Heisenberg, uncertainty paper, 1927
h
"px # "x $
2
!
!
Particles are waves and we can only determine a probability
that the particle is in a certain region of space. We cannot say
exactly where it is!
!
Similarly since E = hf
"E # "t $
h
2
19
20
!
Electron Interference
!
Wave function
!
superposition of 2 waves with same frequency and amplitude
D = D1 + D2 = Asin(kr1"#t) + Asin(kr2 " #t)
!
!
P(x)dx =
Intensity on screen and probability of detecting
electron are connected
=
Computer
simulation
x
!
!
$ "dx '
I(x) = C cos 2 &
% #L )(
!
!
I"A
When doing a light interference experiment, the
probability that photons fall in one of the strips around
x of width dx is
2
!
!
N(in dx at x) Energy(in dx at x) /t
"
=
N tot
N tot hf /t
I(x)Hdx
2
" A(x) dx
N tot hf
The probability of detecting a photon at a particular
point is directly proportional to the square of lightwave 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
photograph
H
22
x
!
Where is most propably the
electron?
your HW
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
x
P(x min " x " x max ) =
!
$
max
x min
2
#(x) 23dx
How do I get a? from
normalization condition
area of triangle =(a x 4)/2 = 1
Suppose now that a is another number, a = 1/2, what is
the probability that the electron is between 0 and 2?
A. 1/2
B. 1/4
C. 2
Probability = area of tringle between 0,2 = (a x 2)/2 =
1/2
24
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
25