Schrodinger’s
Equation
Wave Equation
sin sin 2 x t sin
A x vt A A kx t
T
The general form of the wave function is
which also describes a wave moving in the x direction. In general the amplitude may also be complex.
The wave function is also not restricted to being real. Notice that the sine term has an imaginary number. Only the physically measurable quantities must be real. These include the probability, momentum and energy.
Conditions on the Wave function
In order for to be a solution of the Schrödinger equation to represent a physically observable system, must satisfy certain constraints:
1. Must be a single-valued function of x and t;
2. Must be normalisable; This implies that the 0 as x ;
3. (x) must be a continuous function of x;
Normalization
The probability of finding a particle somewhere in space must be unity, thus the normalization condition:
In 1D: - a wavefunction which obeys this condition is said to be normalized
2,
1
x t
dx
Suppose we have a solution to the Sch. Eq. that is not normalized. The recipe for normalization:
Calculate the normalization integral
Re-scale the wave function as
2
( , ) d
N
x t
x
1
' ,
r t
r t
,
N
This procedure works because any solution of the S.Eq. being multiplied by a constant remains a solution: the S..Eq. is
linear & homogeneous.
r
,
t
2dv
1
The time-dependent Schrödinger equation
2, 2
p
E K U U x t
m
assume, by analogy, that this is a solution of the equation we are looking for
2
2
2
,
,
x t p
x t x
2 2
2
, ,
/ , / , ,
2
x t x t
i x t x t U x t
t m x
2 2
2
, ,
, ,
2
x t x t
i U x t x t
t m x
- the time-dependent Schrödinger Equation in one dimension
x,t =
Ae
-i (t-x/v) For particle moving freely in +x direction=2, v= x,t =
Ae
-i2 (t-x/)E=h=2
; =h/p=2ħ/p
x
,
t
Ae
i EtpxIt represents a wave equivalent of an unrestricted particle of total energy E & momentum p moving in the +x direction
The time-dependent Schrödinger
equation
) t , x ( U m
2 p E
2
For a particle in a potential U (x,t) then
and we have (KE + PE) wavefunction = (Total energy) wavefunction
t i
) t , x ( U x
m
2 2
2 2
TDSE
Points of note:
1. The TDSE is one of the postulates of quantum mechanics. Though the SE cannot be derived, it has been shown to be consistent with all experiments.
2. SE is first order with respect to time (as classical wave equation).
Schrodinger Time independent Equation
Suppose that the potential is independent of time and depends only on position. Therefore we can represent wave function as product of two functions (one dependent on x and another one on t)
2 2 2 2 d xU x x E x
m dx
the time-independent S. Eq in one-dimension.
The solutions of the t-independent S. Eq. corresponds to specific value of energy called the Eigen values and the wave functions are called eigen functions
They correspond to the states that do not change with time (the energy is constant, all average values of all observables are time-independent, etc.).
Solution to the full TDSE is: (x,t)
(x)T(t)
(x)eiEt/From TDSE :
U
x
,
t
x
)
t
,
x
(
m
2
t
)
t
,
x
(
i
2 2 2
Substituting in above equation & dividing whole equation by : / iEt
e
)
x
(
)
t
,
x
(
/ iEte
x
(
x
)
U
x
)
x
(
m
2
)
x
(
E
2 2 2
E
U
0
m
Stationary states
/
) ( )
( ) ( )
,
(x t x T t x eiEt
even when the potential is independent of time the wavefunction still oscillates in time:
Solution to the full TDSE is:
But probability distribution is static:
2 /
/ 2
)
(
)
(
)
(
*
)
,
(
)
,
(
x
t
x
t
x
e
x
e
x
P
iEt
iEt
Particle in a box
Schrödinger Time independent Equation within
the box
A Particle with mass m is restricted to move along x-axis between x=0 and x=L by infinitely hard walls.
Total energy of the particle remains constant as there is no loss of energy due to collision with the walls
Potential energy U is infinite on both sides of the box, while it is constant –(say zero for convenience)- inside the box.
Since particle cannot have infinite energy therefore it cannot exist outside the box.
The particle is thus confined inside the box so that:
=0 for x ≤ 0 and x ≥L
E
U
0
m
2
x
2 2 2
Since is a function of x only thus it partial derivative can be replaced by total derivative
E
U
0
m
2
dx
d
2 2
2
U=0 U(x)
U=∞ U=∞
The solution for schrodinger equation :
(x) = 0 for x 0 and x L
thus: (0) = A sin (0) + Bcos (0)
then : B= 0 from the first boundary condition
We are left with: (x) = Asin(kx)
(L) = A sin (kL) = 0 second boundary condition
If A= 0 we don't have a wave. So sin (kL) must be 0 True for kL = , 2, 3, ..., n
True for k = (/L), (2/L), (3/L), ..., (n/L)
Thus energy can have only certain values for the particle (which are the eigen values) constituting the energy levels of the system
k=(n/L)
We do not include k = 0 because in that case the wave would be 0 for any x.
kx
cos
B
kx
sin
A
mE 2 k
where
2 2 2 2 n
n
mL 2 n E
L n mE
2
Wave functions of a particle whose energies are En with B=0,
x
mE
2
sin
A
nn
L
x
n
sin
A
n
For EnTo normalize we must assign a value to A such that
Pdx dx
2 n
Using
1
dx
dx
2 L
0
n 2
n
A=√(2/L)
Normalized wave functions of the particle are therefore
L
x
n
sin
L
2
n
Contd..
In every case at x=0 and x=L 2 0
n
At a particular place in the box the probability of the particle present may be very different for different quantum numbers (i.e.,n)
