The document derives the time-dependent Schrödinger equation from first principles. It considers a complex plane wave and the Hamiltonian of a system as the sum of kinetic and potential energy. Taking derivatives and multiplying terms leads to the time-dependent Schrödinger equation. The key points are: 1) The Schrödinger equation relates the wavefunction to the total energy of a system as the sum of kinetic and potential energy terms. 2) It is first order in time, involving the complex number i, so solutions are complex rather than real-valued like classical waves. 3) For a free particle, the momentum operator is defined from the derivative of the wavefunction with respect

1. Lecture plan
 Schrödinger equation

2. Derivation of Schrӧdinger’s Wave Equation
Let us consider a complex plane wave:
The Hamiltonian of a system is
where V potential energy & T kinetic energy.
‘H’ is the total energy, we can rewrite the equation as:
Now taking the derivatives,
(3)
(1)
(2)

3. Derivation of Schrӧdinger’s Wave Equation
where ‘λ’ is the wavelength and ‘k’ is the wavenumber
Therefore, k can be written as
Hence, (3) implies
Now, on multiplying Ψ (x, t) to the Hamiltonian (1)
we get,
Now, we know that
(4)
(5)

4. Derivation of Schrӧdinger’s Wave Equation
Now, we know that the energy wave of a matter wave is written
as
Therefore,
Using (4), expression (5) can be written as:
Now, on combining the RHS of (6) & (7) , we can get the
Schrodinger Wave Equation
This is the derivation of Schrödinger Wave Equation
(time-dependent).
t
i
t
x
t
x
E












 )
,
(
)
,
(
(6)
(7)
[Using (2)]

5. Schrӧdinger’s Wave Equation
)
,
( t
x
V
m
2
p
E
2


For a particle in a potential V (x,t) then
and we have (KE + PE)  wavefunction = (Total energy)  wavefunction
t
i
t
x
V
x
m 








 

)
,
(
2 2
2
2
TDSE
Points of note:
1. The TDSE is one of the postulates of quantum mechanics. It has been shown to
be consistent with all experiments.
2. SE is first order with respect to time (cf. classical wave equation).
3. SE involves the complex number i and so its solutions are essentially complex.
This is different from classical waves

6. One dimensional Schrӧdinger’s Wave Equation:
:
)
,
( t
x
 Wave function
:
)
(x
V Potential function
:
m Mass of the particle
t
t
x
t
x
x
V
x
t
x
m
2 2
2
2









 )
,
(
i
)
,
(
)
(
)
,
(


dx
t
x
2
)
,
(
 , the probability to find a particle in (x, x+dx) at time t
2
)
,
( t
x
 , the probability density at location x and time t
Schrӧdinger’s Wave Equation

7. 7
 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:
With k = p / ħ we have
This yields
 This suggests we define the momentum operator as .
 The expectation value of the momentum is
Momentum Operator

8. 2
2
classical
p
H V
m
 
Hamiltonian kinetic potential
energy energy
Sum of kinetic energy and potential energy.
Time dependent Schrödinger Equation
( , , , )
( , , , ) ( , , , )
x y z t
i H x y z t x y z t
t



 
The potential, V, makes one problem different from another. H atom, harmonic oscillator.
Q.M. 2 2
2
(x)
2m x
H V



 
2
2
( , , )
2
H V x y z
m

  
2 2 2
2
2 2 2
x y z
  
  
   
one dimension
three dimensions
p i
x

 

recall
Schrӧdinger’s Wave Equation

9. Next Lecture plan
 We will continue with Schrodinger Equation