Download Translational Motion of a Particle in a Box

Application of Quantum Mechanics:
Translational Motion of a Particle in a Box
For L of same order
of magnitude as
Particle
trapped in
1-D box
Boundary condition:
Particle wave must fit
into the box exactly
h
λ= :
p
Ÿ
2
λ = 2 L, L, L,...
3
2
or λ = L
n
n = 1, 2, 3,...
Nils Walter: Chem 260
The Permitted Wavefunctions of a
Particle in a Box
Ÿ Permitted wavefunctions are:
nπx
Ψn = N sin
; n = 1, 2,...
L
N=
2
L
normalization constant (for scaling)
n=6
L
³
According to Born’s interpretation: Ψ dx = 1
n=5
nπx
2 1
Ÿ N sin
dx = 1 = N
L
³0
L
2
L
2
n=4
n=3
n=2
n=1
0
2
L
L
1
sin 2ax
b/c of ³ sin ( ax ) dx = x −
2
4a 0
0
2
Nils Walter: Chem 260
2
The Permitted (Quantized!) Energies of
a Particle in a Box
n=7
n=6
n=5
n=4
n=3
n=2
h nh
; n = 1, 2,...
According to de Broglie: p = =
λ 2L
p2
and Etotal = Ekin =
n is a quantum
2m
number specifying the
n2h2
Ÿ En =
; n = 1, 2,... energy of the particle
2
8mL
The same result is obtained when applying the
Schrödinger equation to the wavefunction:
2 d 2Ψ
nπx
−
+ VΨ = EΨ; Ψn = N sin
2
2m dx
L
n=1
Only the n=1 state (lowest energy)
Nils Walter: Chem 260
has no nodes; state n has n-1
nodes
E
The Behavior of a Particle in a Box
n=7; 49E1
n=6; 36E1
∆E = En +1 − En
n2h2
En =
; n = 1, 2,...
2
2
h
8mL
= (2n + 1)
8mL2
L small
n=5; 25E1
n=4; 16E1
n=3; 9E1
n=2; 4E1
n=1; E1
Examples:
• electron in NH3(l)
• conjugated polyenes
L large
Zero-point energy (in accord with
Nils Walter: Chem 260
Heisenberg’s uncertainty principle!)
What about a Particle Travelling on a Ring?
Linear momentum p → angular momentum J
J = pr = mvr
Boundary condition:
m↑
↑, v↑
↑ŸJ↑
2
2
1
p
J
=
Etotal = Ekin = mv 2 =
2
2m 2mr 2
hr
moment of inertia I
J = pr =
λ
de Broglie
2πr
Ÿλ=
; n = 0, 1,...
unacceptable
n
acceptable
(
nh / 2π )
=
2
Ÿ En
2I
Nils Walter: Chem 260
n 2 2
=
2I
n = 0, ±1, ±2,...