PHYS 3313 – Section 001
Lecture #16
Wednesday, April 1, 2015
Dr. Jaehoon Yu
•
•
•
Probability of Particle
Schrodinger Wave Equation and Solutions
Normalization and Probability
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
1
Special Project #4
• Prove that the wave function =A[sin(kx t)+icos(kx- t)] is a good solution for the timedependent Schrödinger wave equation. Do NOT
use the exponential expression of the wave
function. (10 points)
• Determine whether or not the wave function
=Ae- |x| satisfy the time-dependent Schrödinger
wave equation. (10 points)
• Due for this special project is Wednesday, Apr. 8.
• You MUST have your own answers!
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
2
The Schrödinger Wave Equation
• Erwin Schrödinger and Werner Heinsenberg
proposed quantum theory in 1920
• The two proposed very different forms of equations
• Heinserberg: Matrix based framework
• Schrödinger: Wave mechanics, similar to the
classical wave equation
• Paul Dirac and Schrödinger later on proved that
the two give identical results
• The probabilistic nature of quantum theory is
contradictory to the direct cause and effect seen
in classical physics and makes it difficult to grasp!
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
3
The Time-dependent Schrödinger Wave Equation
•
The Schrödinger wave equation in its time-dependent
form for a particle of energy E moving in a potential V in
one dimension is
2
¶Y ( x,t )
¶2 Y ( x,t )
i
=+ VY ( x,t )
2
¶t
2m ¶x
•
The extension into three dimensions is
2
æ ¶2 Y ¶2 Y ¶2 Y ö
¶Y
i
=+ 2 + 2 ÷ + VY ( x, y, z,t )
2
ç
¶t
2m è ¶x
¶y
¶z ø
• where i = -1 is an imaginary number
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
4
Ex 6.1: Wave equation and Superposition
The wave equation must be linear so that we can use the superposition principle. Prove
that the wave function in Schrodinger equation is linear by showing that it is satisfied for
the wave equation  (x,t1 (x,t2 (x,t where a and b are constants and 1
(x,t and 2 (x,t describe two waves each satisfying the Schrodinger Eq.
2
¶Y1
¶ 2 Y1
Y = aY1 + bY2
i
=+ VY1
2
¶t
2m ¶x
¶Y ¶
¶Y
¶Y 2
= ( aY1 + bY 2 ) = a 1 + b
¶t ¶t
¶t
¶t
¶Y ¶
¶Y
¶Y 2
= ( aY1 + bY 2 ) = a 1 + b
¶x ¶x
¶x
¶x
2
¶Y
¶2 Y
i
=+ VY
2
¶t
2m ¶x
Rearrange terms
2
¶Y 2
¶2 Y 2
i
=+ VY 2
2
¶t
2m ¶x
¶2 Y ¶ æ ¶Y1
¶Y 2 ö
¶2 Y1
¶2 Y 2
= ça
+b
+b
÷ =a
¶x 2 ¶x è ¶x
¶x ø
¶x 2
¶x 2
2
2
æ ¶
ö
¶Y
¶2 Y
¶2
i
+
VY
=
i
+
V
çè ¶t 2m ¶x 2
÷ø Y = 0
¶t 2m ¶x 2
2
æ ¶ 2 Y1
¶Y
¶Y 2 ö
¶2 Y 2 ö
æ ¶Y1
i
= i ça
+b
a
+b
+ V ( aY1 + bY 2 )
÷ =è ¶t
¶t
¶t ø
2m çè ¶x 2
¶x 2 ÷ø
2
2
æ ¶Y1
ö
æ ¶Y 2
ö
¶ 2 Y1
¶2 Y 2
aç i
+
VY
=
-b
i
+
VY
1÷
2÷ = 0
çè
¶t
2m ¶x 2
è ¶t 2m ¶x 2
ø
ø
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
5
General Solution of the Schrödinger
Wave Equation
• The general form of the solution of the Schrödinger wave
equation is given by:
Y ( x,t ) = Ae
i( kx-w t )
= A éë cos ( kx - w t ) + isin ( kx - w t ) ùû
• which also describes a wave propagating in the x direction. In
general the amplitude may also be complex. This is called the
wave function of the particle.
• The wave function is also not restricted to being real. Only the
physically measurable quantities (or observables) must be
real. These include the probability, momentum and energy.
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
6
Ex 6.2: Solution for Wave Equation
Show that Aei(kx- t) satisfies the time-dependent Schrodinger wave Eq.
i( kx-w t )
¶Y ¶
=
Aei( kx-w t ) = -iAw ei( kx-w t ) = -iwY
¶t ¶t
(
Y = Ae
)
¶Y ¶
=
Aei( kx-w t ) = iAkei( kx-w t ) = ikY
¶x ¶x
(
)
¶2 Y ¶
¶
i( kx-w t )
2 i( kx-w t )
2
=
ikY
=
ik
Y
=
ik
iAke
=
-Ak
e
=
-k
Y
(
)
(
)
2
¶x
¶x
¶x
2 2
æ
ö
k
2
¶Y
w
V
Y=0
2
ç
÷
i
= i ( -iwY ) = wY = -k
Y
+
VY
( )
2m
è
ø
¶t
2m
æ
ö
p2
æw ö
=
E
V
=0
The Energy: E = hf = h ç ÷ = w
ç
÷
2m
è
ø
è 2p ø
2p 2p 2p p p
The momentum: p = k
The wave number: k =
=
=
=
l h p
h
2
2
(
)
p
From the energy conservation: E = K + V =
+V
2m
p
E-V = 0
2m
So Aei(kx- t) is a good solution and satisfies Schrodinger Eq.
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
7
Ex 6.3: Bad Solution for Wave Equation
Determine  (x,tAsin(kx- t) is an acceptable solution for the timedependent Schrodinger wave Eq.
Y = Asin ( kx - w t )
¶Y ¶
= ( Asin ( kx - w t )) = -Aw cos ( kx - w t )
¶t ¶t
¶Y ¶
= ( Asin ( kx - w t )) = kAcos ( kx - w t )
¶x ¶x
¶2 Y ¶
2
2
=
kAcos
kx
w
t
=
-k
Asin
kx
w
t
=
-k
Y
(
)
(
)
(
)
2
¶x
¶x
2
i ( -w cos ( kx - w t )) = -k 2 sin ( kx - w t )) + V sin ( kx - w t )
(
2m
æ 2k2
ö
-i w cos ( kx - w t ) = ç
+ V ÷ sin ( kx - w t )
è 2m
ø
æ p2
ö
-iE cos ( kx - w t ) = ç
+ V ÷ sin ( kx - w t )
è 2m
ø
This is not true in all x and t. So  (x,t)=Asin(kx- t) is not an acceptable
solution for Schrodinger Eq.
Wednesday, April 1, 2015
PHYS 3313-001, Spring 2015
Dr. Jaehoon Yu
8