# The Quantum Mechanics of Simple Systems

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The Quantum Mechanics of Simple Systems
Chemistry 330 The Quantum Mechanics of Simple Systems

Properties of an Acceptable Wavefunction
The wavefunction must be Continuous Single-valued No singularities Continuous first derivatives

The Curvature of a Wavefunction
The average kinetic energy of a particle can be ‘determined’ by noting its average curvature. The observed kinetic energy of a particle is an average of contributions from the entire space covered by the wavefunction. Sharply curved regions contribute a high kinetic energy to the average; slightly curved regions contribute only a small kinetic energy.

The Wavefunction The wavefunction  is a probability amplitude
Square modulus (* or 2) is a probability density. The probability of finding a particle in the region dx located at x is proportional to 2 dx.

Particle Wavefunctions
The wavefunction for a particle at a well-defined location is a sharply spiked function Zero amplitude everywhere except at the particle's position.

Postulates of quantum Mechanics
There exists a wavefunction that is the solution of the Schrödinger equation * - probability density function

<A> - the expectation value of
Postulates #2 The expectation value of any observable is defined as follows <A> - the expectation value of the operator A

- the Hamiltonian operator
Postulates #3 The wavefunction must satisfy the relationship - the Hamiltonian operator

The Spatial and Temporal Functions
Consider the complete wavefunction ((x,y,z,t)) to be a product of a Spatial function – (x,y,z) Temporal function – f(t)

The Hamiltonian and the energy
The eigenvalues for the Hamiltonian operator are the total energy of the system The temporal function describes the variation of the potential energy with time

Commutators and Expectations Values
Two operators that commute Observables corresponding to those operators can have precise values simultaneously Two operators that don’t commute Observables corresponding to those operators can’t be determined simultaneously

An operator that satisfies this condition
Hermitian Operators For an operator An operator that satisfies this condition is said to be Hermitian

Superposition and Expectation Values
A wavefunction that is written as a linear combination The probability of measuring a particular eigenvalue  {cn}2

Superposition and Wavefunctions
The wavefunction for a particle with an ill-defined location Superposition of several wavefunctions of definite wavelength An infinite number of waves is needed to construct the wavefunction of a perfectly localized particle. These wavefunctions interfere constructively in one place but destructively elsewhere. As more waves are used in the superposition (as given by the numbers attached to the curves), the location becomes more precise at the expense of uncertainty in the particle's momentum.

The General Approach Solve the Schrödinger equation for the physical description of the system Obtain expectation values for the observables Obtain the probability density function at various points in space

The Free Particle The particle moves in the absence of an external force x Choose V = 0

The Schrödinger Equation
The wavefunctions

The Particle in a ‘Box’ A particle in a one-dimensional region with impenetrable walls. Potential energy is zero between x = 0 and x = L, Rises sharply to infinity at the walls.

The Schrödinger Equation
The wavefunctions n – energy level L – length of box

The Energy Expression The energy of the particle depends directly on the value of n!

The Solutions for the Particle in a ‘Box’
The first five normalized wavefunctions of a particle in a box. Successive functions possess one more half wave and a shorter wavelength.

The Particle in a ‘Box’ The allowed energy levels for a particle in a box. Note that the energy levels increase as n2, and that their separation increases as the quantum number increases.

The Probability Distributions
The first two wavefunctions and the corresponding probability distributions The probability distribution in terms of the darkness of shading.

Orthogonal wavefunctions
A graphical illustration of orthogonality for two wavefunctions The integral is equal to the total area beneath the graph of the product, and is zero.

Tunnelling Suppose that the energy at the walls dos not rise abruptly to infinity!! A particle incident on a barrier from the left has an oscillating wavefunction, but inside the barrier there are no oscillations (for EV). If the barrier is not too thick, the wavefunction is nonzero at its opposite face, and so oscillations begin again there. (Only the real component of the wavefunction is shown.) V

Tunneling Probability
The probability that the particle will tunnel through the wall Transmission probability Tunneling is dependent on the thickness of the barrier and the mass of the particle. It is important for electrons and muons, less so for protons, and not important for heavy particles. Speed of proton transfer reactions – dependent on tunneling.

The Particle in a 2D ‘Box’
A two-dimensional square well. Potential energy is zero between x = 0 and x = L1 and y= 0 and y = L2, Rises sharply to infinity at the walls. Note that the particle is confined to the surface.

The Schrödinger Equation
The wavefunctions

The Energy Expression The energy of the particle depends directly on the values of nx and ny and ny !

The Quantum Mechanical Harmonic Oscillator
Examine a particle undergoing harmonic motion.

The Schrödinger Equation
The wavefunctions Hv- Hermite Polynomials v – vibrational quantum number

The Energy Expression The energy of the oscillator depends
v – vibrational quantum number The force constant – k Particle mass – m

The First Wavefunction of the QM Oscillator
The normalized wavefunction and probability distribution (shown also by shading) for the lowest energy state of a harmonic oscillator.

Harmonic Oscillator The normalized wavefunction and probability distribution (shown also by shading) for the first excited state of a harmonic oscillator.

The Wavefunctions The first five normalized wavefunctions of the QM harmonic oscillator The normalized wavefunctions for the first five states of a harmonic oscillator. Even values of v are black; odd values are green. Note that the number of nodes is equal to v and that alternate wavefunctions are symmetrical or antisymmetrical about y = 0 (zero displacement).

The Probability Distributions
The probability distributions for the first five states of a harmonic oscillator Note – regions of highest probability move towards the turning points of the classical motion as v increases.

Angular Momentum of A Particle Confined to a Plane
Represented by a vector of length |ml| units along the z-axis Orientation that indicates the direction of motion of the particle. The direction is given by the right-hand screw rule.

Quantization of Rotation
Examine a particle undergoing rotation in a plane

The Schrödinger Equation

The Wavefunctions The wavefunctions are dependent on the quantum number ml

The Momenta and their Operators
The angular momentum operators are written as follows Eigenvalue - Jz Eigenvalue – J 2

Wavefunctions of the Particle on a Ring
The real parts of the wavefunctions of a particle on a ring. For shorter wavelengths, the magnitude of the angular momentum around the z-axis grows in steps of ħ.

3-D Rotation Suppose we allow the particle to move on the surface of a sphere. Two angles  - the azimuthal angle  - the colatitude

The Schrödinger Equation
2 – the Laplacian Operator

The Wavefunctions The wavefunctions are dependent on the angles  and . The Schrödinger equation is simplified by the separation of variables technique.

The Solutions The solutions to the SE for this systems are the spherical harmonics l ml Yl,ml 1/(4)1/2 1 3/(4)1/2 Cos ±1 -+ 3/(8)1/2 Sin e±i

The Probability Distributions
A representation of the wavefunctions of a particle on the surface of a sphere. Note that the number of nodes increases as the value of l increases. All these wavefunctions correspond to ml = 0; a path around the vertical z-axis of the sphere does not cut through any nodes.

Space Quantization Represent the vectors for the angular momenta as a series of cones!

The Stern-Gerlach Experiment
The experimental arrangement for the Stern-Gerlach experiment: the magnet provides an inhomogeneous field. The classically expected result. The observed outcome using silver atoms.

The Spin Functions of Electrons
An electron spin (s = 1/2) can take only two orientations with respect to a specified axis.  electron (top) - electron with ms = +1/2;  electron (bottom) is an electron with ms = - 1/2. The vector representing the magnitude of the spin angular momentum lies at an angle of 55 to the z-axis (more precisely, at arccos (1/3 1/2)).