“Anyone who can contemplate quantum mechanics without getting dizzy hasn’t understood it ” Neils Bohr Discussion Is there a particle flying faster than light speed ? Lecture 10 Particles in 3D Potenti[.]
Trang 1“Anyone who can contemplate
quantum mechanics without getting dizzy hasn’t understood it.”
Neils Bohr Discussion: Is there a particle flying faster than light speed ?
Trang 2Lecture 10: Particles in 3D Potentials and the Hydrogen
Atom
0 0.5
1 1
0 g( ) x
4
P(r)
0
1
r
r = a0
z
x
L
L L
2 2 2 2
2
n
n
mL
h E
z
y
) ( ) ( ) ( )
, ,
( x y z x y z
3 o
e a
1 )
r
2
6 13
n
eV
E n
Trang 3 3-Dimensional Potential Well
Product Wavefunctions
Concept of degeneracy
Early Models of the Hydrogen Atom
Planetary Model
Quantum Modifications
Schrödinger’s Equation for the Hydrogen Atom
Ground state solution
Spherically-symmetric excited states (“s-states”)
Trang 4Quantum Particles in 3D Potentials
One consequence of confining a quantum particle in two or three dimensions is “degeneracy” the occurrence of
several quantum states at the same energy level.
So far, we have considered quantum particles
bound in one-dimensional potentials This
situation can be applicable to certain physical
systems but it lacks some of the features of
many “real” 3D quantum systems , such as
atoms and artificial quantum structures:
(www.kfa-juelich.de/isi/)
A real (3D)
“quantum dot”
To illustrate this important point in a simple system, we
extend our favorite potential the infinite square well
to three dimensions.
Trang 5Particle in a 3D Box (1)
outside box, x or y or z < 0
outside box, x or y or z > L
0 inside box U(x,y,z) =
Let’s solve this SEQ for the particle in a 3D box:
y
The extension of the Schrödinger Equation (SEQ) to 3D
is straightforward in cartesian (x,y,z) coordinates:
E )
z , y , x (
U dz
d dy
d dx
d
2
2 2
2 2
2 2
2
) , , ( x y z
where
This simple U(x,y,z) can be “separated”
U(x,y,z) = U(x) + U(y) + U(z)
z
x
L
L L
Kinetic energy term in the Schrödinger Equation
2 2 2
2
1
z y
p m like
Trang 6Particle in a 3D Box (2)
So, the whole problem simplifies into three one-dimensional
equations that we’ve already solved in Lecture 7.
) ( )
( ) ( )
(
2 2
x E
x x
U dx
x d
m x
) ( )
( ) (
) (
2 2
y E
y y
U dy
y d
m y
L
n N
n
( ) sin
2 2
2
L
n m
h
L
n N
n
( ) sin
2 2
2
L
n m
h
Likewise for ( z)
So the Schrödinger Equation becomes:
E )
z ( U ) y ( U ) x (
U dz
d dy
d dx
d
2
2 2
2 2
2 2
2
and the wavefunction can be “separated” into the product of three functions: ( x , y , z )
) ( ) ( ) ( )
, ,
graphic
Trang 7Particle in a 3D Box (3)
So, finally, the eigenstates and associated
x
y
L
L
L
L
n y
L
n x
L
n
sin sin sin
where n x ,n y , and n z can each have values 1,2,3,….
This problem illustrates 2 important new points.
(1) Three ‘quantum numbers’ (n x ,n y ,n z ) are needed to completely identify the state of this three-dimensional system.
(2) More than one state can have the same energy:
“Degeneracy”.
Degeneracy reflects an underlying symmetry in U(x,y,z)
3 equivalent directions
2 2 2
2
2
n n
mL
h E
z y
Trang 8Consider a particle in a two-dimensional (infinite) well, with Lx = L y
1 Compare the energies of the (2,2), (1,3), and (3,1) states?
a E (2,2) > E (1,3) = E (3,1)
b E (2,2) = E (1,3) = E (3,1)
c E (1,3) = E (3,1) > E (2,2)
2 If we squeeze the box in the x-direction (i.e., L x < L y ) compare
E (1,3) with E (3,1) :
a E (1,3) < E (3,1)
b E (1,3) = E (3,1)
c E (1,3) > E (3,1)
Lecture 10, exercise 1
Trang 9Consider a particle in a two-dimensional (infinite) well, with Lx = L y
1 Compare the energies of the (2,2), (1,3), and (3,1) states?
a E (2,2) > E (1,3) = E (3,1)
b E (2,2) = E (1,3) = E (3,1)
c E (1,3) = E (3,1) > E (2,2)
2 If we squeeze the box in the x-direction (i.e., L x < L y ) compare
E (1,3) with E (3,1) :
a E (1,3) < E (3,1)
b E (1,3) = E (3,1)
c E (1,3) > E (3,1)
Lecture 10, exercise 1
E (1,3) = E (1,3) = E 0 (1 2 + 3 2 ) = 10 E 0
The tighter confinement along x will increase the contribution to E The effect will be
greatest on states with greatest n x :
E (2,2) = E 0 (2 2 + 2 2 ) = 8 E 0
2
8
x y
y
h
mL
Example: L x = L y /2
2 2
2
8
x y
y x
n n
n n
h E
m L L
Trang 10Energy levels (1)
Now back to a 3D cubic box:
Show energies and label (n x ,n y ,n z ) for the first
11 states of the particle in the 3D box, and
write the degeneracy D for each allowed energy
Use E o = h 2 /8mL 2
z
x
y
L
L L
E