Relativity
EM waves &
PE effect
Matterwaves
Schroedinger
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Relativity
What does “proper time” and
“ proper length” refer to?
Observer A measures the velocity of a rocket as v,
and a comet as u, both traveling in the same
direction. What is the speed of the comet as
measured by Observer B on the rocket?
u
u  v 
b)  u  v 
c)  u  v 
d)  u  v 
e)  u  v 
a)
2
1

uv
c


2
2
1

v
c


1  v2 c2
2
1  uv c 
2
1

uv
c


S’
v
S
Application of the Space-time interval
Two events occur in the frame S.
Event 1 (x1 = -0.5s*c, t1=1s)
Event 2: (x2 = 0, t2 = 2s)
What’s the proper time between these two events?
(s)2 = (ct)2 - (x)2= (c*1s)2 – (0.5*c)2 = 0.75 c2
A) 0 s
Proper time: x’ ≡ 0
B) 0.25 s
 0.75 c2 = (c* tproper)2 – (0)2
C) 0.5 s
 tproper = 0.87 s
D) 0.75s
E) None of the above
Velocity transformation
A high-speed train is traveling at a velocity of v = 0.5c. The
moment it passes over a bridge it launches a cannon ball
straight up (as seen by the train conductor) with a velocity of
0.4c. What is the velocity of the ball right after it was
launched as seen by an observer standing on the bridge?
Attach reference frame S to the train:
Observer is in frame S' traveling from right to left (v is negative!!)
ux = 0
Now use the velocity
y
y'
uy = 0.4c
transformation:
ux  v
u' x 
1  ux v / c 2
S
S'
uy
v = -0.5c
u' y 
x
x'
2
 1  ux v / c 
Velocity transformation
}
u'x = 0.5c
u'y = 0.346c
Velocity transf.
ux  v
u' x 
1  ux v / c 2
uy
u' y 
 1  ux v / c 2 
u'  (u' x )2  (u' y )2  0.61c
ux = 0
uy = 0.4c
y
S
x
y'
S'
v = -0.5c
x'
Lucy
v
?
... -3
-2
-1
0
1
George
2
3 ...
x   ( x  vt)
v
t    (t  2 x)
c
George has a set of synchronized clocks in reference
frame S, as shown. Lucy is moving to the right past
George, and has (naturally) her own set of synchronized
clocks. Lucy passes George at the event (0,0) in both
frames. An observer in George’s frame checks the clock
marked ‘?’. Compared to George’s clocks, this one reads
A) a slightly earlier time
B) a slightly later time
C) same time
Lucy
x   ( x  vt)
v
t    (t  2 x)
c
v
?
... -3
-2
-1
0
1
George
2
3 ...
The event has coordinates (x = -3, t = 0) for George.
In Lucy’s frame, where the ? clock is, the time t’ is
v
3 v
t    (0  2 ( 3))  2
c
c
, a positive quantity.
‘?’ = slightly later time
EM & PE effect
How could you generate light with an
electron?
a.
b.
c.
d.
e.
Stationary charges
Charges moving at a constant velocity
Accelerating charges
b and c
a, b, and c
Stationary charges 
constant E-field, no magnetic (B)-field
E
+
Charges moving at a constant velocity 
Constant current through wire creates a B-field
But B-field is constant
I
Accelerated charges 
changing E-field and changing B-field
B
(EM radiation  both E and B are oscillating)
B
E
EM radiation often represented by a sinusoidal curve.
OR
What does that sinusoidal curve tell you?
What stuff is moving up & down in space?
How to get population inversion in
this two-level system?
e
You can tweak color and intensity
ΔE
e
excited
not excited
An electron bashes into an atom in a discharge lamp
Electron leaves hot filament with
nearly zero initial kinetic energy
- V +
Batt
-2 eV
-3 eV
-8 eV
-9 eV
Atom is fixed at the center of the tube.
How does the voltage VBatt (4 < VBatt < 10 V)
influence the color of this discharge lamp?
Case 1:
0V
e-
distance: d
Case 2:
1V
+
+
+
+
0V
e-
d/2
1V
+
+
+
+
Electron in both cases initially at rest.
Which electron has higher final kinetic energy just
before it hits the right plate?
In the photoelectric effect: what can
you say about the influence of the
battery voltage on the work function Φ?
Initial KE vs. f:
I
Current vs. Voltage:
hf = 
0
high intensity
low intensity
U
Initial KE
(voltage independent)
0
hfmin=
Frequency
Matter waves
(x,t=0)
(x,t=0)= x/L from -L to L
(x,t=0)=0 elsewhere

dx
a
-L
b
0
L
x
How do the probabilities of finding the electron
described by Ψ(x,t) above near of a and b compare
(within an interval dx)?
In the deBroglie picture, the electrons have an
intrinsic wavelength associated with them.
We also know that one wavelength fits around
the circumference for the n=1 level of hydrogen,
…, 5 fit for n=5, etc.
Does that mean that the n=5 circumference is 5
times as large as the n=1 circumference?
Explain!
What’s the value of ‘a’?
x)
ψ(|x|>L)=0
a/3
L
-L
X
-a
ψ(x) & Ψ(x,t): traveling or
standing waves?
 2  2 ( x)

 V ( x) ( x)  E ( x)
2
2m x
 2  2 ( x, t )
( x, t )

 V ( x )( x, t )  i
2
2m x
t
What is a superposition state?
Could you explain it based on this special case?
( x, t )  1 / 2  1 ( x, t )  1 / 2  2 ( x, t )
where:
Ψ1(x,t)=1(x)e–iE1t/ and
Ψ2(x,t)=2(x)e–iE2t/ are the ground state and
first excited state of the infinite square well.
Schrödinger
Name two aspects of the
hydrogen atom Bohr got wrong!
What does this equation describe?
   ( x)

 E ( x)
2
2m x
2
2
To solve the hydrogen atom, what
are the first two steps you take?
2 2

  ( r,  ,  )  V ( r,  ,  ) ( r,  ,  )  E ( r,  ,  )
2m
For the finite quantum well: name three
changes that would increase how far the wave
penetrates into the classically forbidden region!
 ( x)  Be
x
2m
(V  E )
with:  
2

0
L
If the total energy E of the electron is
LESS than the work function of the metal,
V0, when the electron reaches the end of
the wire, it will…
Write down and proof the “grand
unified theory” (and publish it)!