The figure shows a potential energy function.

(a) How much energy could a classical particle have and still be bound?

(b) Where would an unbound particle have its maximum kinetic energy?

(c) For what range of energies might a classical particle be bound in either of two different regions?

(d) Do you think that a quantum mechanical particle with energy in the range referred to in part?

(e) Would be bound in one region or the other? Explain.

(a)For a particle to be bound, its total energy must be less than the potential energy function at points in a space to either side of the particle.

This potential energy function approaches 7J as positions, x, approaches infinity.

$\underset{\mathrm{x}\to \infty }{\lim }\mathrm{U}\left(\mathrm{x}\right)=7\mathrm{J}$

The potential energy function approaches infinity as position, x, approaches 0.

$\underset{\mathrm{x}\to \infty }{\lim }\mathrm{U}\left(\mathrm{x}\right)=\infty$

Any particle with less than 7J energy will be bound because the potential energy will be greater to either side.

The total amount of energy that the particle has, E, is conserved. Total energy is the sum of the kinetic energy, KE, and the potential energy, U.

E = KE+U

Solve the expression of total energy, E, for kinetic energy, KE.

KE = K - U

Because total energy is constant, the maximum kinetic energy occurs when potential energy reaches its maximum.

In this case that is at x=0.1

Therefore, the maximum kinetic energy of unbound particle have at x=0.1

A particle reaches a turning point when it’s total energy, E, is equal to its potential energy, U.

If a particle with a specific energy can be bound in two different regions, then there will be more than ² turning points.

To find the energy range that produces two bound regions, draw horizontal lines representing total energy, E , on the graph of potential energy, U, and look for points of intersection. Total energy lines that intersect the potential energy curve in 3 or more places will produce particles that can be confined to two different regions

Therefore, the particles with energy between 4J and 5J will be bound in two distinct regions.

In quantum mechanics, the wave function is solved for specific regions, specifically ones where the potential energy is higher than the total energy of the particle, and ones where the potential energy is lower than the total energy of the particle.

When a particle has more total energy than potential energy in a given region, the result is a bound state.

When a particle has less total energy than potential energy in a given region, the wave function decays exponentially in space. This is called tunnelling.

Hence, a quantum mechanical particle would not be confined to one region in the energy range of 4J to5J because it would be able to tunnel through the potential energy barrier to the other region.