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The Schrödinger Wave Equation
The Schrödinger Wave Equation
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
The Schrödinger Wave Eq.
Classical Mechanics : Wave Mechanics
= Geometrical optics: Wave optics
classical mechanics
quantum mechanics
classical mechanics ←→ Newton’s theory = geometrical optics
wave (quantum) mechanics ←→ Huygen’s theory = wave optics
quantum phenomena ←→ diffraction & interference
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Time-independent Schrödinger Wave Eq.
wave
1  2  (r , t )
eq.： (r, t )  2
,
v
 t2
2
solution is assumed to be sinusoidal,
→ (r, t )   (r ) exp( -i t )
→ Helmholtz eq.  2  k 2  (r )  0
de Broglie relation k  p /  → k  2m [ E  V (r)] 
Erwin Schrödinger
2
→ time-indep. Schrödinger eq.：     2  V (r)  (r)  E (r)


 2m

the appearance of  → Schrödinger imposed the “quantum condition”
on the wave eq. of matter
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Time-dependent Schrödinger Wave Eq.
Einstein relation： E   also represents the particle energy
 E 
t



→ (r, t )   (r ) exp  -i
Schrödinger found a 1st-order derivative in time consistent with the
time-indep. Schrödinger eq.
 E 
(r, t )   (r ) exp  -i t 
  
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
The Probability Interpretation
the probability density of finding the particle ：
 (r, t )  (r, t )
2
wave function = field distribution
its modulus square = probability density distribution
∵the particle must be somewhere, ∴total integrated = 1

(r, t ) d 3 r    (r, t )d 3 r  1
2
(the wave function for the probability interpretation needs to be
normalized. )
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
The Probability Interpretation
N identically particles, all described by  (r, t )
the number of particles found in the interval (r, r  dr ) at t ：
2
dN (r, t )  N  (r, t ) d 3 r  N (r, t ) d 3 r
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
The Probability Current Density
a time variation of  (r, t ) in a region is conserved by a net change in flux
into the region.
→ J (r , t ) satisfy a continuity eq. ：
  (r, t )
t
   J (r, t )  0
by analogy with charge conservation in electrodynamics,
  (r, t )
t
J(r, t ) 
   J (r, t )  0
←→ conservation of probability

 (r, t )(r, t )  (r, t ) (r, t )
2im


2006 Quantum Mechanics

Prof. Y. F. Chen
The Schrödinger Wave Equation
Role of the Phase of the Wave Function
J (r , t ) is related to the phase gradient of the wave function.
 (r, t )   (r, t ) exp[ i(r, t )]
, where (r, t )  tan 1  Im[ (r, t )]  = phase of the wave function
 Re[ (r, t )] 
→
  (r, t )  (r, t )   (r, t )   (r, t )  i (r, t ) (r, t )
 (r, t )   (r, t )   (r, t )   (r, t )  i (r, t ) (r, t )
→ J(r, t )   (r, t )

m
(r, t )
the larger (r, t ) varies with space, the greater J (r , t )
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Role of the Phase of the Wave Function
J(r, t )   (r, t )

m
(r, t )
reveals that the J (r , t ) is irrotational only
when (r, t ) has no any singularities, which are the points
of Re[ (r, t )]  Im[ (r, t )]  0 .
Conversely, the singularities of (r, t ) play a role of vortices to cause
J (r , t )
to be rotational.
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Wave Functions in Coordinate and Momentum Spaces
with p  k
 (r , t ) 
1
( 2) 3 / 2
 (p, t ) 
1
(2) 3 / 2

i

 ( p, t ) exp  p  r  d 3 p



 i

(r, t ) exp   p  r  d 3 r
 

normalized：  (r, t ) d 3 r   (p, t ) d 3 p  1
2
2
 (p, t ) d 3 p is the probability of finding the momentum of the particle in
2
d 3 p in the neighborhood of p at time t
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Operators and Expectation values of Physical Variables
expectation value of r & p：
 r   r (r, t ) d 3 r
2
,
 p   p (p, t ) d 3 p
2
find an expression for <p> in coordinate space：

 p    (r, t )  i
   (r, t ) d 3r
→ <p> can be represented by the differential operator p̂  i 
any function of p , f (p) & any function of r , g (r ) can be given by：

 g (r )  g (r )

 f (p) 
2006 Quantum Mechanics

d r   (p, t ) g (i  ) (p, t ) d

f (p)  (p, t ) d 3 p 
2
(r, t )
2
3
  (r, t ) f ( i ) (r, t ) d 3 r

p
3
p
Prof. Y. F. Chen
The Schrödinger Wave Equation
Operators and Expectation values of Physical Variables
CM：all physical quantities can be expressed in terms of coordinates &
momenta.
QM： all physical quantities F (r,p) can be given by


3
  (r, t ) F (r,  i )  (r, t ) d r

 F (r, p)  


3
  (p, t ) F (i  p , p) (p, t ) d p



any physical operator in quantum mechanics needs to a Hermitian
operator.
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Time Evolution of Expectation values & Ehrenfest’s Theorem
the operators used in QM needs to be consistent with the requirement
that their expectation values generally satisfy the laws of CM
the time derivative of x can be given by： d  r   J (r,t ) d 3 r
dt

→ integration by parts：
d r 1
p

  (r,t ) i    (r,t ) d 3 r 
dt
m
m

→ the classical relation between velocity and p holds for the
expectation values of wave packets.
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Time Evolution of Expectation values & Ehrenfest’s Theorem
Ehrenfes’s theorem：the time derivative of p can be given by
d p
  V (r )  F(r ) 
dt
→ has a form like Newton’s 2nd law, written for expectation values
for any operator A, the time derivative of <A> can be given by：
i
(1) where Ĥ
A
d  A
 AH  HA  i
dt
t
2 2
ˆ
is the Hamiltonian operator H     V (r )
2m
(2) the eq. is of the extreme importance for time evolution of
expectation values in QM
2006 Quantum Mechanics
Prof. Y. F. Chen
The Schrödinger Wave Equation
Stationary States & General Solutions of the Schrödinger Eq.
superposition of eigenstates：based on the separation of t & r
(r, 0)   cE  E (r)
&
E
E
(r, t )   cE  E (r) exp   i t 
  
E
(1) E = eigenvalue
(2)  E (r) = eigenfunction
 2 2
(3)     V (r)  E (r)  E E (r)
 2m

stationary states：if the initial state is represented by  E (r)
→ (r, t )   E (r) exp   i t 

E
 
→  (r, t )  (r, t ) 2   E (r) 2 , independent of t
2006 Quantum Mechanics
Prof. Y. F. Chen