ACER 9/16 Lecture #5: Wavefunctions and Particle in a Box

Write the Schrödinger equation.

Eψ(x) = Ĥ ψ(x)

• Hamiltonian operator Ĥ is the total energy of the system. It can be used to determine “all but the finest details of atoms and molecules”.

What is the Schrödinger equation used for?

These scenarios are…

used to map out complex chemistry starting from simple scenarios like a particle in a box, which could then extend to an electron in an atom/bond (e.g. an electron trapped in the orbital of a hydrogen atom), then to vibrating molecules, and so on.

These scenarios are stationary states and their solutions are time-independent solutions. This means the probability of where you can find a particle is frozen in time. This does not mean that the particle itself is frozen still.

What’s a Limitation of the Schrödinger Equation?

Wavefunction ψ cannot be measured directly so we can’t check if our interpretation of a scenario is correct. Instead, we measure the probability density.

What is Probability Density (how is it calculated)?

What is Born Interpretation and where does it apply?

|ψ(x)²|

Born Interpretation is the rule that probability density gives the probability of finding an electron at a certain position x within the chosen state—the energy level being observed—in one dimension (for this form of probability density).

What does “Normalizing” the wavefunction mean? What mathematical expression can represent this?

Since the particle must be somewhere within the chosen state, the wavefunction can be normalized by adjusting the math so that the total probability of finding the particle somewhere in the space is exactly 1 (100%).

Describe the Particle in a Box Scenario and its boundary conditions.

Discuss energy (include what specific energy and its notation).

A particle in a 1-dimensional box of length L approximates an electron in a bond.

Boundary conditions are set:

• Inside the box (between x=0 and x=L), the potential energy of the system is 0. V(x) = 0

  • This means the only energy inside the box is the kinetic energy of the particle moving around.

• At the walls and outside of the box, the potential energy is infinite. V(x) = ∞

  • This means that ψ (x) = 0 outside the box. The electron can only exist inside the box.

Describe the solution to the particle in a box problem (two equations).

These equations tell you…

The solution to this problem describes the quantized/discrete energy levels of the particle.

What is a Node?

How do you determine the number of nodes that are in a wavefunction or probability density graph?

a position in the box where wavefunction is zero (ψ = 0), so the particle cannot exist at that point (probability density will also be zero ψ² = 0)

The number of nodes is the number of the energy level minus 1 (n - 1).

Draw a wavefunction ψ graph at energy levels n = 1 and n = 2.

Draw a probability density |ψ|² graph at energy levels n = 1 and n = 2.

For a probability density graph of n = 2, describe: What is the probability of the particle being in either half of the box? What is the probability of the particle existing in the very middle of the box?

The probability of the particle being in either half of the box is 50%.

The probability of the particle being in the very middle of the box is 0%. There is a node at x = L/2 so the probability density there is zero.

What is Zero-Point Energy? Why does it exist (connect this to another concept)?

Discuss uncertainty.

the lowest achievable energy in the system, where n = 1 (ground state)

E = 0 is not possible since the particle is never perfectly still, satisfying the uncertainty principle (The particle is trapped so its position uncertainty is restricted to the width of the box. If the particle were still then its momentum is also known which is not possible.).

• Uncertainty in x can never be greater than L. The possibilities where the particle can be cannot be greater than the box itself.