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Final Exam
Tuesday, May 8, 2012
Starting at 8:30 a.m.,
Hoyt Hall.
Summary of Chapter 38
•  In Quantum Mechanics particles are represented
by wave functions Ψ.
•  The absolute square of the wave function |Ψ|2
represents the probability of finding particle in a
given location.
•  The wave function Ψ satisfies the
Time-Dependent Schrödinger Equation
⎡  ∂
⎤
∂
⎢ − 2m ∂x 2 + V (x) ⎥ Ψ(x,t) = i ∂t Ψ(x,t)
⎣
⎦
2
2
Time-Dependent Schrödinger Equation
⎡  2 ∂2
⎤
∂
⎢ − 2m ∂x 2 + V (x) ⎥ Ψ(x,t) = i ∂t Ψ(x,t)
⎣
⎦
Time-Independent Schrödinger Equation
⎧ 2 d 2
⎪ −
ψ (x) + V (x)ψ (x) = Eψ (x)
2
⎪ 2m dx
⎨
d
1
⎪
f (t) = Ef (t)
⎪⎩
dt
i
E
→ Ψ(x,t) = ψ (x)exp[−i t]

Summary of Chapter 38
•  A particle in an infinite potential well has
quantized energy levels:
•  In a finite well, probability extends into
classically forbidden areas.
•  Particles can tunnel through barriers of
finite height and width.
38.7 Free Particles; Plane Waves and
Wave Packets
Example 38-4: Free electron.
An electron with energy E = 6.3 eV is in free
space (where V = 0). Find
(a) the wavelength λ (in nm) and
(b) the wave function for the electron
(assuming B = 0).
38.7 Free Particles; Plane Waves and
Wave Packets
The solution for a free
particle is a plane wave, as
shown in part (a) of the
figure; more realistic is a
wave packet, as shown in
part (b). The wave packet
has both a range of
momenta and a finite
uncertainty in width.
38.8 Particle in an Infinitely Deep Square
Well Potential (a Rigid Box)
One of the few potentials
where the Schrödinger
equation can be solved
exactly is the infinitely
deep square well.
As is shown, this
potential is zero from the
origin to a distance ,
and is infinite elsewhere.
38.8 Particle in an Infinitely Deep Square
Well Potential (a Rigid Box)
The solution for the region between the
walls is that of a free particle:
Requiring that ψ = 0 at x = 0 and x =

gives B = 0 and k = nπ/ .
This means that the energy is limited to
the values:
2
h
En = n
;
2
8ml
2
n = 1, 2, 3, ...
38.8 Particle in an Infinitely Deep Square
Well Potential (a Rigid Box)
The wave function for each of the quantum states is:
ψ(x) = Asin kn x ;
π
kn = n
l
The constant A is determined by the condition
+∞
∫
2
ψ(x) dx = 1;
−∞
+∞
∫
−∞
2
Asin kn x dx = A
l
2
∫ sin kn x dx =1;
0
2
2
ψn =
sin kn x
l
38.8 Particle in an Infinitely Deep Square
Well Potential (a Rigid Box)
These plots show for several values of n the energy
levels, wave function, and probability distribution.
energy levels
wave function
probability distribution
Simulations
38.8 Particle in an Infinitely Deep Square
Well Potential (a Rigid Box)
Example 38-5: Electron in an infinite potential
well.
(a)  Calculate the three lowest energy levels for
an electron trapped in an infinitely deep
square well potential of width  = 0.1 nm
(about the diameter of a hydrogen atom in
its ground state).
(b) If a photon were emitted when the electron
jumps from the n = 2 state to the n = 1 state,
what would its wavelength be?
38.8 Particle in an Infinitely Deep Square
Well Potential (a Rigid Box)
Example 38-7: Probability near center of
rigid box.
An electron is in an infinitely deep square
well potential of width  = 1.0 x 10-10 m. If
the electron is in the ground state, what
is the probability of finding it in a region
dx = 1.0 x 10-12 m of width at the center of
the well (at x = 0.5 x 10-10 m)?
38.10 Tunneling Through a Barrier
Since the wave function
does not go to zero
immediately upon
encountering a finite
barrier, there is some
probability of finding
the particle represented
by the wave function on
the other side of the
barrier. This is called
tunneling.
Figure 38-15 goes
here.
38.10 Tunneling Through a Barrier
The probability that a particle tunnels
through a barrier can be expressed as a
transmission coefficient, T, and a reflection
coefficient, R (where T + R = 1). If T is small,
The smaller E is with respect to U0, the
smaller the probability that the particle will
tunnel through the barrier.
38.10 Tunneling Through a Barrier
Example 38-11: Barrier penetration.
A 50-eV electron approaches a square
barrier 70 eV high and (a) 1.0 nm thick,
(b) 0.10 nm thick. What is the probability
that the electron will tunnel through?
Chapter 39
Quantum Mechanics of
Atoms
Units of Chapter 39
•  Quantum-Mechanical View of Atoms
•  Hydrogen Atom: Schrödinger Equation and
Quantum Numbers
•  Hydrogen Atom Wave Functions
•  Complex Atoms; the Exclusion Principle
•  The Periodic Table of Elements
•  X-Ray Spectra and Atomic Number
•  Magnetic Dipole Moments; Total Angular
Momentum
39.1 Quantum-Mechanical View of
Atoms
Since we cannot say exactly where an electron
is, the Bohr picture of the atom, with electrons
in neat orbits, cannot be correct.
Quantum theory
describes an electron
probability distribution:
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
Potential energy for the hydrogen atom:
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
The time-independent Schrödinger
equation in three dimensions is then:
where
37-11 The Bohr Model
In each of its orbits, the
electron would have a
definite energy:
Z2
En = −(13.6 eV) 2 ,
n
n = 1, 2, 3,...
37-10 Atomic Spectra: Key to the
Structure of the Atom
A portion of the complete spectrum of hydrogen
is shown here. The lines cannot be explained by
the Rutherford theory.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
There are four different quantum numbers
needed to specify the state of an electron in
an atom.
1.  The principal quantum number n gives the
total energy.
2.  The orbital quantum number  gives the
angular momentum;  can take on integer
values from 0 to n - 1.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
3. The magnetic quantum number, m,
gives the direction of the electron’s
angular momentum, and can take on
integer values from –  to + .
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
This plot indicates the
quantization of angular
momentum direction
for  = 2.
The other two
components of the
angular momentum are
not defined.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
In a magnetic field, the energy levels
split depending on m (Zeeman effect).
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
4. The spin quantum number, ms, for an
electron can take on the values +½ and -½.
The need for this quantum number was found by
experiment; spin is an intrinsically quantum
mechanical quantity, although it mathematically
behaves as a form of angular momentum.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
This table summarizes the four quantum
numbers.