HOMEWORK SOLUTIONS

WEEK 4 PROBLEMS

 

Ch. 5

#16.) An electron is trapped in an infinitely deep potential well 0.300 nm in width.

(a)   If the electron is in its ground state, what is the probability of finding it within 0.100 nm of the left hand wall?

 

The ground state wavefunction for the infinite well is

 

     y(x) = Ö(2/L)sin(px/L)

 

 

where L = 0.300 nm.  Therefore, the probability to find the electron in the given range is

 

     P = ò|y(x)|²dx

 

        = (2/L)ò₀^0.100sin²(px/L)dx

 

        = 0.196

 

 

(b)   Repeat (a) for an electron in the 99^th energy state above the ground state.

 

The wavefunction in this case is

 

     y(x) = Ö(2/L)sin(99px/L)

 

The probability integral is similar to the one above

 

     P = ò|y(x)|²dx

 

        = (2/L)ò₀^0.100sin²(99px/L)dx

 

        = 0.333

 

 

(c)   Are your answers consistent with the correspondence principle?

 

The correspondence principle says that for large quantum numbers (like, oh, I don’t know… n=99), the quantum result should be identical to the result you would obtain doing it the classical way.  So what would be the classical answer in this problem?  Well, the box is 0.300 nm long and we are looking at the probability to find the electron in the range 0 to 0.100 nm… in other words, the first 1/3 of the box.  Classically, the probability to find the particle is the same at any point in the box (see problem #29 in the homework set), meaning that the probability to find the particle in the first third of the box is

 

     P = (1/3) = 0.333

    

 

So, for n=99, the answer is approaching the classical value.