Q3 (32 points) Figure 2 shows a potential function with incident particles coming from -o∞ with a tota Vị < E < V2. i. Let the wavenumbers in regions I, II and III be k1, k2, and kg. Provide expressions for k1,/ (6 points). ii. What are the time-independent wave equations V(x) in regions I, II and III (6 points). iii. Write down the possible boundary conditions at r = 0, r = a and r > a. Do not solv constants of the wave functions (16 points). Incident particles V < E < V2 V2 II III

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Q3 (32 points)
Figure 2 shows a potential function with incident particles coming from -o with a total energy
Vị < E < V2.
i. Let the wavenumbers in regions I, II and III be k1, k2, and kg. Provide expressions for k1, k2 and k3
(6 points).
ii. What are the time-independent wave equations V(x) in regions I, II and III (6 points).
iii. Write down the possible boundary conditions at r = 0, x = a and r > a. Do not solve for the
constants of the wave functions (16 points).
Incident particles V, <E <V2
V2
V1
I
II
III
x= 0
X= a
Figure 2. Potential function for Q3.
2. Prove that the Fermi-Dirac probability at E = E, is half (4 points).
Transcribed Image Text:Q3 (32 points) Figure 2 shows a potential function with incident particles coming from -o with a total energy Vị < E < V2. i. Let the wavenumbers in regions I, II and III be k1, k2, and kg. Provide expressions for k1, k2 and k3 (6 points). ii. What are the time-independent wave equations V(x) in regions I, II and III (6 points). iii. Write down the possible boundary conditions at r = 0, x = a and r > a. Do not solve for the constants of the wave functions (16 points). Incident particles V, <E <V2 V2 V1 I II III x= 0 X= a Figure 2. Potential function for Q3. 2. Prove that the Fermi-Dirac probability at E = E, is half (4 points).
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