PHYS 3313 – Section 001
Lecture #13
Wednesday, Oct. 23, 2013
Dr. Jaehoon Yu
•
•
•
•
•
Wave Function Normalization
Time-Independent Schrödinger Wave
Equation
Expectation Values
Operators – Position, Momentum and
Energy
Infinite Square Well Potential
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
1
Announcements
• Mid-term grade discussion Monday, Oct. 28
–
–
–
–
In Dr. Yu’s office, CPB 342
Last name begins with A – C: 12:50 – 1:20pm
Last name begins with D – L: 1:20 – 1:50pm
Last name begins with M – Z: 1:50 – 2:20pm
• Colloquium today
– 4pm today, SH101, Dr. X. Chu, U. of Colorado
– Double extra credit for this colloquium
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
2
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
3
Ex 6.4: Normalization
Consider a wave packet formed by using the wave function that Ae- ,
where A is a constant to be determined by normalization. Normalize this
wave function and find the probabilities of the particle being between 0 and
1/ , and between 1/ and 2/ .
Y = Ae
Probabilit
y density
ò
+¥
-¥
-a x
Y Ydx = ò
*
-¥
+¥
+¥
-¥
0
= ò A2 e-2a x dx = 2 ò
A= a
Wednesday, Oct. 23, 2013
+¥
( Ae ) ( Ae )dx = ò ( A e )( Ae )dx =
-a x
*
+¥
-a x
* -a x
-a x
-¥
2
2A -2a x
2 -2a x
e
Ae
dx =
-2a
Normalized Wave Function
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
+¥
= 0+
0
A2
a
= 1
Y = ae
-a x
4
Ex 6.4: Normalization, cont’d
Using the wave function, we can compute the probability for a particle to be
with 0 to 1/ and 1/ to 2/ .
Y = ae
-a x
For 0 to 1/ :
P=ò
1a
0
Y Ydx = ò
*
1a
0
a -2a x
-2a x
e
a e dx =
-2a
1a
0
1 -2
= - ( e -1) » 0.432
2
For 1/ to 2/ :
P=ò
2a
1a
Y Ydx =
*
2a
ò a ae
1
2a
-2a x
a -2a x
1 -4 -2
e
=
e - e ) » 0.059
dx =
(
-2a
2
1a
How about 2/ :to ∞?
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
5
Properties of Valid Wave Functions
Boundary conditions
To avoid infinite probabilities, the wave function must be finite
everywhere.
2) To avoid multiple values of the probability, the wave function must be
single valued.
3) For finite potentials, the wave function and its derivatives must be
continuous. This is required because the second-order derivative
term in the wave equation must be single valued. (There are
exceptions to this rule when V is infinite.)
4) In order to normalize the wave functions, they must approach zero as
x approaches infinity.
Solutions that do not satisfy these properties do not generally
correspond to physically realizable circumstances.
1)
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
6
Time-Independent Schrödinger Wave Equation
• The potential in many cases will not depend explicitly on time.
• The dependence on time and position can then be separated
in the Schrödinger wave equation. Let, Y ( x,t ) = y ( x ) f ( t )
2
¶ f (t )
f ( t ) ¶2y ( x )
which yields: i y ( x )
=+ V ( x )y ( x ) f ( t )
2
¶t
2m
¶x
2
1 ¶ f (t )
1 ¶2y ( x )
i
=+ V ( x)
2
f ( t ) ¶t
2m y ( x ) ¶x
Now divide by the wave function:
• The left side of this last equation depends only on time, and
the right side depends only on spatial coordinates. Hence each
side must be equal to a constant. The time dependent side is
1 df
i
=B
f dt
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
7
Time-Independent Schrödinger Wave Equation(con’t)

We integrate both sides and find: i
df
ò f = ò Bdt Þ i ln f = Bt + C
where C is an integration constant that we may choose to be 0.
Bt
Therefore
ln f =
i
This determines f to be by comparing it to the wave function of a free
particle
f ( t ) = eBt i = e-iBt = e-iw t Þ B = w Þ B = w = E
1 ¶ f (t )
i
=E
f ( t ) ¶t

This is known as the time-independent Schrödinger wave
equation, and it is a fundamental equation in quantum mechanics.
d 2y ( x )
+ V ( x )y ( x ) = Ey ( x )
2
2m dx
2
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
8
Stationary State
• Recalling the separation of variables: Y ( x,t ) = y ( x ) f ( t )
-iw t
and with f(t) = e
the wave function can be
-iw t
written as:
Y ( x,t ) = y ( x ) e
• The probability density becomes:
Y Y=y
*
2
( x )( e
iw t -iw t
e
) =y ( x )
2
• The probability distributions are constant in time.
This is a standing wave phenomena that is called the
stationary state.
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
9
Comparison of Classical and
Quantum Mechanics
•
•
•
Newton’s second law and Schrödinger’s wave
equation are both differential equations.
Newton’s second law can be derived from the
Schrödinger wave equation, so the latter is the more
fundamental.
Classical mechanics only appears to be more precise
because it deals with macroscopic phenomena. The
underlying uncertainties in macroscopic measurements
are just too small to be significant due to the small size
of the Planck’s constant
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
10
Expectation Values
• In quantum mechanics, measurements can only be expressed in
terms of average behaviors since precision measurement of each
event is impossible (what principle is this?)
• The expectation value is the expected result of the average of many
measurements of a given quantity. The expectation value of x is
denoted by <x>.
• Any measurable quantity for which we can calculate the expectation
value is called a physical observable. The expectation values of
physical observables (for example, position, linear momentum, angular
momentum, and energy) must be real, because the experimental
results of measurements are real.
N i xi
å
N
x
+
N
x
+
N
x
+
N
x
+
• The average value of x is x = 1 1
2 2
3 3
4 4
= i
N1 + N 2 + N 3 + N 4 +
åN
i
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
11
i
•
Continuous Expectation Values
+¥
We can change from discrete to
xP ( x ) dx
ò
-¥
continuous variables by using
the probability P(x,t) of
observing the particle at a
particular x.
• Using the wave function, the x =
expectation value is:
• The expectation value of any
function g(x) for a normalized
wave function:
+¥
x=
ò
+¥
-¥
+¥
ò
ò
xY ( x,t ) Y ( x,t ) dx
-¥
+¥
-¥
P ( x ) dx
*
Y ( x,t ) Y ( x,t ) dx
*
g ( x ) = ò Y ( x,t ) g ( x ) Y ( x,t ) dx
*
-¥
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
12
Momentum Operator
• To find the expectation value of p, we first need to represent p in
terms of x and t. Consider the derivative of the wave function of a
free particle with respect to x:
¶Y ¶ i( kx-w t )
éë e
ùû = ikei( kx-w t ) = ikY
=
¶x ¶x
With k = p / ħ we have ¶Y = i p Y
¶x
¶Y ( x,t )
This yields
p éë Y ( x,t ) ùû = -i
¶x
• This suggests we define the momentum operator as
• The expectation value of the momentum is
p =
pˆ = -i.
¶
¶x
¶Y ( x,t )
ò-¥ Y ( x,t )pˆ Y ( x,t ) dx = -i ò-¥ Y ( x,t ) ¶x dx
+¥
+¥
*
Wednesday, Oct. 23, 2013
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
*
13
Position and Energy Operators


The position x is its own operator as seen above.
The time derivative of the free-particle wave function
is
¶ i( kx-w t )
¶Y
i( kx-w t )
¶t
éë e
¶t
=
Substituting 


ùû = -iw e
= -iwY
¶Y ( x,t )
= E / ħ yields E éë Y ( x,t ) ùû = i
¶t
¶
ˆ
E=i
¶t
The energy operator is
The expectation value of the energy is
E =
ò
+¥
-¥
ˆ ( x,t ) dx = i
Y ( x,t )EY
Wednesday, Oct. 23, 2013
*
PHYS 3313-001, Fall 2013
Dr. Jaehoon Yu
¶Y ( x,t )
ò-¥ Y ( x,t ) ¶t dx
+¥
*
14