The graph shows the potential energy for a one-dimensional quantum system. For e
ID: 2077008 • Letter: T
Question
The graph shows the potential energy for a one-dimensional quantum system. For each of the following ranges of energy, determine whether possible wavefunctions exist with energy in the range. If so, determine whether the possible energies of the wavefunctions will be continuous or discrete and if the wavefunctions will be bound or unbound. Explain. Assume that E+ and E- are the top and bottom asymptotes of the potential respectively.
1. Range E+ < V(x)
2. Range E- < V(x) < E+
3. Range E0 < V(x) < E-
E+ E- E0 EE VExplanation / Answer
1. Range E+ < V(x) :
We will have a wave function in the region E+ > V(x), as the wave function must be smooth and continuos , it exists in th region E+ <V(x) too but it will decay soon on the left side, it is like a barrier pentration, from wave mechanics , the particle has certian probability of being in the region. As the wave function is decaying the enrgy is continuous.
The wave function is bound
2. Range E- < V(x) < E+
Similar argument as above, the particle do exist in the region E- > V(x) , as the wave function must be smooth and continuous it do exist in the specified region but it is decaying and with a penetration depth decided by the actual potential and other parameters of the particle. There is some probability of findign the particle in this region and hecne the wave function exist. the enrgies are continuous as the wave function is decaying.
3. Range E0 < V(x) < E-
By the same arguments as above the wave function exists and it is exponentially decaying as the wave do exit for
V(x) > Eo and V(x) less than E- , the particle will have some probability in this region too,
4. Range V(x) < E0
The particle do not exist in any region for this case and there is no wave function do not exist anywhere.
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