Final Exam

Tuesday, May 8, 2012

Starting at 8:30 a.m., Hoyt Hall.

•  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

Summary of Chapter 38

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)

Time-Dependent Schrödinger Equation

−2

2md 2

dx2ψ (x) +V (x)ψ (x) = Eψ (x)

ddtf (t) = 1

iEf (t)

⎧

⎨⎪⎪

⎩⎪⎪

→ Ψ(x,t) =ψ (x)exp[−i Et]

Time-Independent Schrödinger Equation

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 h a s b o t h a r a n g e o f m o m e n t a a n d a f i n i t e 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:

En = n2 h2

8ml2; 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) = Asinknx ; kn = nπl

ψ(x) 2 dx−∞

+∞

∫ =1;

Asinknx2 dx

−∞

+∞

∫ = A 2 sinknx2 dx =

0

l

∫ 1;

The constant A is determined by the condition

ψn =2lsinknx

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

•  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

Units of Chapter 39

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:

39.1 Quantum-Mechanical View of Atoms

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

In each of its orbits, the electron would have a definite energy:

37-11 The Bohr Model

En = −(13.6 eV) Z2

n2, n = 1,2,3,...

A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.

37-10 Atomic Spectra: Key to the Structure of the Atom

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

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).

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.

39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers