Consider the semi-infinite potential, as shown in the figure below, where V(x)=infty for x<=0,V= 0 for V(x)=V_(0)x>=LELcot^(2) heta +1=csc^(2) heta ,csc heta =(1)/(sin heta )0, and V(x)=V_(0) for x>=L.
(a) Starting with the time-independent Schrodinger equation, find an expression for the transcendental equation from which the energies, E, for the bound states can be found.
(b) Assuming there is at least one bound state, sketch what the wave function of the ground state might look like.
(c) Is the energy of the bound ground state higher, lower, or the same as that for the ground state of the infinite square well of width L ? Explain.
(d) Are there conditions for which no bound state is possible? What are those conditions?
(Note: cot^(2) heta +1)=csc^(2) heta ,csc heta =((1)/(sin heta ))
3. Consider the semi-infinite potential, as shown in the figure below, where V(x) = co for x 0, V 7Z x I0J %=(x)4 pue 7>x> 0 i0J 0 (a) Starting with the time-independent Schrodinger equation, find an expression for the transcendental equation from which the energies, E, for the bound states can be found.
(b) Assuming there is at least one bound state, sketch what the wave function of the ground state might look like.
(c) Is the energy of the bound ground state higher, lower, or the same as that for the ground state of the infinite square well of width L? Explain.
(d) Are there conditions for which no bound state is possible? What are those conditions?
(Note: cot20+ 1=csc20,csc0= 1/sin0)
V(x
V
0
X