Homework 7. extra problem (9) added.

Notes: 7 and 8 were added  Feb 25, 8 PM.  Problem 9 (extra credit) added Feb 27, 9 AM. Also, I made 5 move specific and hopefully more clear.

1.  a) Show that \(\psi_k = e^{ikx}\) is an energy eigenstate of the Schrodinger equation for a free electron:
$$ - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x) = E \psi(x) $$ What is its energy eigenvalue, \(E_k\)?



b) $$\Psi(x,t) =\int_{-\infty}^{+\infty} \phi(k)\, e^{ikx} \,e^{-iE_kt/\hbar} dk$$ is a non-specific wave-packet state. Briefly explain the nature of the 3 terms in this integrand, and why the integral is over k.
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Specific case: Let's consider the specific case of: \(\phi(k)= \frac{\sqrt{a}}{(2 \pi)^{1/4}}  e^{-(k-k_o)^2a^2}\).
c) At t=0, integrating the equation above with this specific \(\phi(k)\) leads to:
\(\Psi(x,0) =\frac{1}{(a\sqrt{2 \pi})^{1/2}} e^{-x^2/(4a^2)} e^{ik_o x}\).
Describe this wave-function. Check its normalization.

2. This is a continuation of the previous problem, using the same integral form of \(\Psi(x,t)\) and the same \(\phi (k)\). 
a) For t not equal to zero, integrating over k leads to:

\(\Psi(x,t) =\frac{1}{(a\sqrt{2 \pi} (1+iT))^{1/2}} e^{-(x-2 k_o a^2 T)^2/(4a^2(1+T^2)}e^{ik_o x/(1+T^2)} e^{ix^2T/(4a^2(1+T^2))} e^{-i(k_o a)^2T/(1+T^2)}\)
where  \(T=\frac {\hbar \, t} {2ma^2}\) is a unit-less quantity proportional to time (t). (This definition helps make the result more readable and understandable.)
a) Describe this wave-function or, if you prefer, you can describe \(|{\Psi (x,t)}|^2\) (which gets rid of the phase factors).
b) How far would you say the electron has traveled in time t?
c) What would you say is the electrons speed?
d) How much has the width changed by T=1? What is t at T=1?

3. Consider a 1-dimensional semiconductor doped so that there are \(5 \times 10^4 \)electrons/cm in its conduction band.
a) Suppose that is the presence of an electric field (an externally applied voltage), each electron acquires an average velocity of \(10^5\) cm/sec. What is the current associated with these conduction electrons?  (in coulombs/second)
b) Comment on the size of this current.
c) Just for fun, what is the current in electrons/second?

4. In a crystalline solid such as a semiconductor:
a) Can a filled band carry current?
b) Can an empty band carry current?
c) Can a partially filled band carry current?

5. In a simple semiconductor:
a) Each energy band comes from a particular s----- a--- e----- e--------- ?
b) Including spin, each band has a total number of states of (multiple choice): N/2, N, 2N, 1000
(where N is the number of atoms in the semiconductor).

6. Consider a square well with 5 bound state and 4 electrons. Assume that it is completely okay to ignore electron-electron interactions and we can therefore just put the electrons into single-electron bound states.
a) At low temperature (T=0), what states will be occupied and which will be empty? (Assume spin.)
b) What energy of photon would it take to excite an electron from the 1st excited state to the 2nd excited state?
c) Suppose you arrange 100 of these into a perfect one-dimensional crystal. How many total states would there be in the band associated with the ground state?
d) How many total electrons would you have in this 1D crystal?
e) What bands would be occupied and and which would be empty (T=0).
f) Roughly what energy of photon would it take to excite an electron from the highest occupied crystal state to the lowest unoccupied crystal state?

7. part I. Consider a square well with 5 bound state and 4 electrons. Suppose you arrange 100,000 of these into a perfect one-dimensional crystal.
a) How many total states would there be in the conduction band?
b) How many total states would there be in the valence band?
c) Suppose that you mix in 0.1% of atoms that are similar, but have 5 electrons. That is, you have 99.9% 4-electron atoms and 0.1% with 5 electrons.  How many electrons would there be in the conduction band?

7. part II. Now suppose that you mix in 0.1% of atoms that are similar, but have 3 electrons. That is, you have 99.9% 4-electron atoms and 0.1% with 3 electrons. 
d) How many electrons would there be in the valence band?
e) How many empty (unoccupied) states would there be in the valence band?
f)  How many holes would there be in the valence band?

8. Suppose you had an entire crystal made of 5 electron atoms (square wells as above), arranged in a perfect 1D array of 100,000 atoms.
a) How many electron would there be in the conduction band in this case?
(This is not a semi-conductor.)

9. (extra credit, late add) This is a question I should have asked, but I didn't think of it. It is not difficult, but provides useful background for understanding n-p junctions (which is why i though f it now).  Here is a chance to get some easy extra-credit.
a) When you dope a semi-conductor so that it has electrons in the conduction band, would the semi-conductor be negatively charged or would it be charge neutral? If you answer is "charge neutral", how can that be (since it has a number of negatively charged electrons in the CB)? Explain?
b) When you dope a semi-conductor so that it has holes in the valence band, would the semi-conductor be positively charged or would it be charge neutral? If you answer is "charge neutral", explain how.

11. [You do not have to do this problem. It is totally optional. This is a statistical mechanics problem involving finite temperature. It is peripheral, not central, to what we will be covering. Comment here on whether you are interested in this or not.]
a) Have you heard of the Fermi function? I think it is \( f(E) = \frac {1} {e^{(E-E_F )/KT} +1}\)
b) Suppose \(E_F\) is exactly half way in between the energy of the 1st and 2nd excited states of the square well system of problem 6. Additionally suppose the \(E_3 - E_2 = 1\) eV and that KT =25 meV (room temperature). What is the probability, in a stat mech sense, of a thermal electron in the second excited state?
c) In a crystal of 1 billion square wells, how many (thermal) electrons would there be in the 2nd excited state band? How many empty states would there be in the lower band? (Assume zero bandwidth and 1 eV between the 1st-excited state band and the 2nd-excited state band to simplify this problem.)
d) Because the band gap is a lot larger than KT, is the exponential term in the denominator of \(f(E)\) a lot bigger than the 1? What approximate expression for \(f(E)\) would you get if you ignore the 1 in the denominator? What is this called? (Boltzman statistics maybe?)