Schrödinger
Equation
Background
It was developed in 1925 by the Austrian physicist
Erwin Schrödinger.
The Schrödinger equation is a fundamental
equation in quantum mechanics that describes the
behavior of particles in terms of waves. In simple terms, it
tells us how the wave function of a particle changes over
time.
The wave function of a particle describes the
probability of finding that particle at a certain location at a
certain time. The Schrödinger equation tells us how this
wave function changes over time, based on the energy of
the particle and the forces acting on it.
The equation takes the form of a partial differential equation,
which can be quite complex, but its solutions provide us with
important information about the behavior of particles at the
quantum level. By solving the Schrödinger equation, we can
calculate the probability of finding a particle in a particular state or
location, and make predictions about how particles will behave in
certain circumstances.
Overall, the Schrödinger equation is a key tool for
understanding the behavior of particles at the quantum level, and
has important applications in variety of fields.
REVIEW
A Partial Derivative is a derivative where we hold some variables constant.
The partial derivative is denoted by the symbol ∂, which replaces the
roman letter d used to denote a full derivative.
The Marquis de Condorcet used this symbol for partial differences in 1770,
which is one of the earliest recorded applications of it in mathematics. AdrienMarie Legendre invented the modern partial derivative notation in 1786, but he
later abandoned it. Carl Gustav Jacob Jacobi revived the symbol in 1841.
Sample Problem
Find
∂f
∂f
and
for
∂x
∂y
Practice Problem 1
Second partial derivatives
Example
Second partial derivatives
Practice Problem 2
Mixed second derivatives
Example
Example
Mixed second derivatives
Practice Problem 3
Find
! !"
!#!$
and
! !"
!$!#
for
HAMILTONIAN OPERATOR
1
𝜕𝜓 (𝑥, 𝑡)
+ 𝜓(𝑥, 𝑡)
𝑖ℏ
=𝐻
𝜕𝑡
KE OPERATOR
2
+ = 𝑇- + 𝑉𝐻
3
ℏ% 𝜕 %
𝑇- = −
2𝑚 𝜕𝑥 %
4
ℏ% 𝜕 %
+=−
𝐻
+ 𝑉%
2𝑚 𝜕𝑥
Substitute eqn 3 in eqn 2
5
𝜕𝜓 (𝑥, 𝑡)
ℏ% 𝜕 %
𝑖ℏ
= −
+ 𝑉- 𝜓(𝑥, 𝑡)
%
𝜕𝑡
2𝑚 𝜕𝑥
Substitute eqn 4 in eqn 1
PE OPERATOR
EXPLICIT EXPRESSION
6
𝜓 𝑥, 𝑡 = 𝜙 𝑥 𝜉(𝑡)
7
𝜕(𝜙 𝑥 𝜉(𝑡))
ℏ% 𝜕 %
𝑖ℏ
= −
+ 𝑉- (𝜙 𝑥 𝜉(𝑡))
%
𝜕𝑡
2𝑚 𝜕𝑥
Separation of variables
Substitute eqn 6 in eqn 5
𝜕(𝜙 𝑥 𝜉(𝑡))
ℏ% 𝜕 %
- 𝑥 𝜉(𝑡)
𝑖ℏ
=−
𝜙 𝑥 𝜉 𝑡 + 𝑉𝜙
%
𝜕𝑡
2𝑚 𝜕𝑥
8
𝜕(𝜉(𝑡))
ℏ% 𝜕 %
- 𝑥 𝜉(𝑡)
𝑖ℏ𝜙 𝑥
= −𝜉(𝑡)
𝜙 𝑥 + 𝑉𝜙
%
𝜕𝑡
2𝑚 𝜕𝑥
rearrange
9
Multiply both sides by
1
𝜙 𝑥 𝜉(𝑡)
𝜕(𝜉(𝑡))
1
ℏ! 𝜕 !
1
0 𝑥 𝜉(𝑡)
𝑖ℏ𝜙 𝑥
=−
𝜉 𝑡
𝜙
𝑥
+
𝑉𝜙
!
𝜙 𝑥 𝜉(𝑡)
𝜕𝑡
𝜙 𝑥 𝜉 𝑡
2𝑚 𝜕𝑥
𝜙 𝑥 𝜉 𝑡
1
1
𝜕(𝜉(𝑡))
1 ℏ% 𝜕 %
𝑖ℏ
=−
𝜙 𝑥 + 𝑉%
𝜉(𝑡)
𝜕𝑡
𝜙 𝑥 2𝑚 𝜕𝑥
10
Separate
𝑖ℏ 𝜕(𝜉(𝑡))
=𝐸
𝜉(𝑡) 𝜕𝑡
ℏ%
𝜕%
−
𝜙 𝑥 + 𝑉- = 𝐸
%
𝜙 𝑥 2𝑚 𝜕𝑥
11
ℏ%
𝜕%
- 𝑥 =𝐸𝜙 𝑥
𝜙 𝑥 −
𝜙 𝑥 + 𝑉𝜙
%
𝜙 𝑥 2𝑚 𝜕𝑥
ℏ% 𝜕 %
- 𝑥 =𝐸𝜙 𝑥
−
𝜙 𝑥 + 𝑉𝜙
%
2𝑚 𝜕𝑥
ℏ% 𝜕 %
−
𝜓 𝑥 + 𝑉- 𝜓 𝑥 = 𝐸 𝜓 𝑥
%
2𝑚 𝜕𝑥
Time-independent Schrödinger
Equation
12
𝑖ℏ 𝜕(𝜉(𝑡))
𝜕𝑡
= 𝐸 𝜕𝑡
𝜉(𝑡) 𝜕𝑡
𝜕(𝜉(𝑡))
𝑖ℏ
= 𝐸 𝜕𝑡
𝜉(𝑡)
13
1 𝜕(𝜉(𝑡)) 1
𝑖ℏ
= 𝐸 𝜕𝑡
𝑖ℏ
𝜉(𝑡)
𝑖ℏ
𝜕(𝜉(𝑡)) 𝐸
=
𝜕𝑡
𝜉(𝑡)
𝑖ℏ
Multiply both sides by
1
𝑖ℏ
14
𝜕(𝜉(𝑡)) 𝑖𝐸
= % 𝜕𝑡
𝜉(𝑡)
𝑖 ℏ
multiply numerator and denominator
by i
𝜕(𝜉(𝑡))
𝑖𝐸
=−
𝜕𝑡
𝜉(𝑡)
ℏ
15
𝜕(𝜉(𝑡))
𝑖𝐸
8
= 8−
𝜕𝑡
𝜉(𝑡)
ℏ
𝑖𝐸𝑡
ln(𝜉(𝑡)) = −
ℏ
Integrate each side
16
𝑒
&'()(*))
=𝑒
TEMPORAL
𝜉(𝑡) = 𝑒
,
,
-.*
ℏ
-.*
ℏ
simplify
SPATIAL
17
𝜓 𝑥, 𝑡 = 𝜙 𝑥 𝜉(𝑡)
𝜓 𝑥, 𝑡 = 𝜙 𝑥 𝑒
,
-.*
ℏ
Substitute the result in eqn 17
18
19
20
ℏ% 𝜕 %
−
𝜓 𝑥 + 𝑉- 𝜓 𝑥 = 𝐸𝜓 𝑥
%
2𝑚 𝜕𝑥
𝑇- 𝜓 𝑥 + 𝑉- 𝜓 𝑥 = 𝐸𝜓 𝑥
+ = 𝐸𝜓
𝐻𝜓
THE HAMILTONIAN OPERATOR THAT ACTS
ON THE WAVEFUNCTION IS EQUAL TO THE
EIGENENERGY TIMES PSI/WAVEFUNCTION
In other words, the Hamiltonian Operator tells you how much
energy a system has and how that energy is distributed among its
constituent particles.
The Hamiltonian operator is used to calculate the time evolution
of a system in quantum mechanics. It acts on the wave function of the
system to determine how it changes over time.
In summary, the Hamiltonian operator is a fundamental concept in
quantum mechanics that helps us understand the behavior and energy
of particles in a system.
QUANTUM INTERPRETATIONS
MANY WORLDS , COPENHAGEN INTERPRETATION,
QUANTUM DECOHERENCE, BOHMIAN MECHANICS