3.012 Fund of Mat Sci: Bonding – Lecture 2
THINK OUT OF THE BOX
@Bobby Douglas, from photo.net
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time: Wave mechanics
1.
2.
3.
4.
5.
6.
Classical harmonic oscillator
Kinetic and potential energy
De Broglie relation λ • p = h
“Plane wave”
Time-dependent Schrödinger’s equation
A free electron satisfies it
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Wed 14
• Study: 15.1, 15.2
• Read: 14.1-14.4
• Office Hours – Monday 4-5 pm
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Time-dependent Schrödinger’s equation
(Newton’s 2nd law for quantum objects)
• An electron is fully described by a wavefunction – all the
properties of the electron can be extracted from it
• The wavefunction is determined by the differential equation
r
r
r
r
h
∂Ψ (r , t )
2
−
∇ Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) = ih
2m
∂t
2
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Stationary Schrödinger’s Equation (I)
r
r
r
r
∂Ψ (r , t )
h
2
−
∇ Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) = ih
2m
∂t
2
*
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Stationary Schrödinger’s Equation (II)
⎡ h
r ⎤ r
r
2
∇ + V (r )⎥ϕ (r ) = Eϕ (r )
⎢−
⎦
⎣ 2m
2
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Stationary Schrödinger’s Equation (III)
⎡ h
r ⎤ r
r
2
∇ + V (r ) ⎥ ϕ (r ) = Eϕ (r )
⎢−
⎣ 2m
⎦
2
1.
It’s not proven – it’s postulated, and it is confirmed experimentally
2.
It’s an “eigenvalue” equation: it has a solution only for certain values (discrete,
or continuum intervals) of E
3.
For those eigenvalues, the solution (“eigenstate”, or “eigenfunction”) is the
complete descriptor of the electron in its equilibrium ground state, in a
potenitial V(r).
4.
As with all differential equations, boundary conditions must be specified
5.
Square modulus of the wavefunction = probability of finding an electron
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
From classical mechanics to operators
• Total energy is T+V (Hamiltonian is kinetic +
potential)
r
p
• classical momentum
→
→ gradient operator
r
r
r
− ih∇
• classical position
→
→ multiplicative operator
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
r̂
Operators, eigenvalues, eigenfunctions
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Free particle: Ψ(x,t)=φ(x)f(t)
h2 2
−
∇ ϕ ( x ) = Eϕ ( x )
2m
d
ih f (t ) = E f (t )
dt
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Infinite Square Well (I)
(particle in a 1-dim box)
h d ϕ ( x)
−
= Eϕ ( x)
2
2m dx
2
2
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Infinite Square Well (II)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Infinite Square Well (III)
20
18
16
Energy
14
12
10
8
6
18
Energy or wave function value
20
16
14
12
10
8
6
4
4
2
2
0
0.0
0.2
0.4
0
0.6
0.8
1.0
x/a
Figure by MIT OCW.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)