Particle in a Box - 1D (14.3)

• A simple problem that demonstrates the principles of quantum mechanics is the particle in a box (PIB)– Inside the box of a given length (a), the potential is zero; outside the box the potential is

infinite– The wavefunction must be zero outside the box since the PE can’t be infinite

• The solution to the SE for the PIB is an oscillating function– It is sinusoidal since the wavefunction must be zero at the ends of the box– An arbitrary integer (n) is in the solution and is referred to as a quantum number

• The energy of the PIB is also dependent on the quantum number– Quantum number means the energy is quantized (only has certain values)– Energy also depends on the length of the box

€

En =h2n2

8ma2

€

ψn x( ) =2

asin

nπx

a

⎛

⎝ ⎜

⎞

⎠ ⎟ n =1, 2, 3, ...

Properties of PIB Wavefunctions (14.3-14.4)

• Each solution of the PIB SE represents a state of the system– The lower the state (smaller n) the lower the energy of the system– First state is referred to as the ground state; higher energy states are called excited

states

• Some interesting features arise from the solution of the PIB SE– As the quantum number increases, so does the number of values of x for which the

PIB wavefunction equals zero (nodes)– The PIB probability densities show a number of interesting characteristics (non-uniform

probabilities, regions of zero probability within the box)

• PIB for higher dimensions (2D, 3D) show very similar behavior– Wavefunction is a product of 1D PIB wavefunctions– Energy is a sum of 1D PIB energies– Some energy levels are degenerate (have same energies)

€

ψnxnyx,y( ) =ψ nx x( )ψ ny x( ) E =

h2

8m

nx2

a2+ny

2

b2

⎛

⎝ ⎜

⎞

⎠ ⎟

• In order for a system to go from one state to another, it must absorb or emit a quantized amount of energy (radiation)– Light is often used to promote particles to higher states (absorption)– Light is often emitted when excited states go back to lower lying states (emission)

• Spectroscopy is the use of light to probe states in matter– Type of light needed depends on the physical problem one is interested in (i.e., the

potential energy function)– Selection rules dictate whether two states can be connected by shining light on it

(depends on the nature of the wavefunctions of the two states)

• Does PIB model any real physical phenomena?– The spectroscopy of highly conjugated molecules (e.g., polyenes) can be explained quite

well by PIB model

Excited States and Spectroscopy (18.1-18.2)

€

hν = E i+1 − E i i =1, 2, 3, ...

Particle in a Box

PIB Wavefunctions

PIB Probability Densities

Absorption and Emission of Radiation