Question B2 Consider a potential barrier with V (x) = Vo cos² (2) for - 1/4 < x < and which is zero everwhere else, as illustrated below. V(x) Vo -L/2 L/2 0 Figure 2: A potential barrier with V(x) = V₁ cos² (7). a) Identify expressions for the values of x at the edges of the "classically forbidden region" (i.e. the region that needs to be tunnelled through), for a particle with energy E < Vo. b) [4 marks] Calculate the leading-order contribution to the Gamow factor for a particle with energy E to tunnel through this barrier, under the assumption that E <<< Vo. N.B. You may wish to refer to the Appendix when answering this question. [8 marks] c) The barrier has height V and width L. Using your result from part b), determine how the transmission rate of particles that tunnel through this barrier would change if (i) the height of the barrier is doubled. (ii) the width of the barrier is doubled. (iii) the barrier is altered so that V = 0 when x > 0. [6 marks] d) Assume that two barriers like the one in Figure 2 are placed back-to-back, and that particles with energy EV are trapped between them. Calculate the half-life 1/2 of a population of such particles. [7 marks]
Question B2 Consider a potential barrier with V (x) = Vo cos² (2) for - 1/4 < x < and which is zero everwhere else, as illustrated below. V(x) Vo -L/2 L/2 0 Figure 2: A potential barrier with V(x) = V₁ cos² (7). a) Identify expressions for the values of x at the edges of the "classically forbidden region" (i.e. the region that needs to be tunnelled through), for a particle with energy E < Vo. b) [4 marks] Calculate the leading-order contribution to the Gamow factor for a particle with energy E to tunnel through this barrier, under the assumption that E <<< Vo. N.B. You may wish to refer to the Appendix when answering this question. [8 marks] c) The barrier has height V and width L. Using your result from part b), determine how the transmission rate of particles that tunnel through this barrier would change if (i) the height of the barrier is doubled. (ii) the width of the barrier is doubled. (iii) the barrier is altered so that V = 0 when x > 0. [6 marks] d) Assume that two barriers like the one in Figure 2 are placed back-to-back, and that particles with energy EV are trapped between them. Calculate the half-life 1/2 of a population of such particles. [7 marks]
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![Question B2
Consider a potential barrier with V (x) = Vo cos² (2) for - 1/4 < x < and which is zero everwhere
else, as illustrated below.
V(x)
Vo
-L/2
L/2
0
Figure 2: A potential barrier with V(x) = V₁ cos² (7).
a) Identify expressions for the values of x at the edges of the "classically forbidden region"
(i.e. the region that needs to be tunnelled through), for a particle with energy E < Vo.
b)
[4 marks]
Calculate the leading-order contribution to the Gamow factor for a particle with energy E to
tunnel through this barrier, under the assumption that E <<< Vo.
N.B. You may wish to refer to the Appendix when answering this question.
[8 marks]
c) The barrier has height V and width L. Using your result from part b), determine how the
transmission rate of particles that tunnel through this barrier would change if
(i) the height of the barrier is doubled.
(ii) the width of the barrier is doubled.
(iii) the barrier is altered so that V = 0 when x > 0.
[6 marks]
d) Assume that two barriers like the one in Figure 2 are placed back-to-back, and that particles
with energy EV are trapped between them. Calculate the half-life 1/2 of a population
of such particles.
[7 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c0d52ac-ebeb-4643-8051-a3a6486fdce4%2F7787344c-fa27-42b4-bc2d-31e33a2a5d68%2F3m3jxt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question B2
Consider a potential barrier with V (x) = Vo cos² (2) for - 1/4 < x < and which is zero everwhere
else, as illustrated below.
V(x)
Vo
-L/2
L/2
0
Figure 2: A potential barrier with V(x) = V₁ cos² (7).
a) Identify expressions for the values of x at the edges of the "classically forbidden region"
(i.e. the region that needs to be tunnelled through), for a particle with energy E < Vo.
b)
[4 marks]
Calculate the leading-order contribution to the Gamow factor for a particle with energy E to
tunnel through this barrier, under the assumption that E <<< Vo.
N.B. You may wish to refer to the Appendix when answering this question.
[8 marks]
c) The barrier has height V and width L. Using your result from part b), determine how the
transmission rate of particles that tunnel through this barrier would change if
(i) the height of the barrier is doubled.
(ii) the width of the barrier is doubled.
(iii) the barrier is altered so that V = 0 when x > 0.
[6 marks]
d) Assume that two barriers like the one in Figure 2 are placed back-to-back, and that particles
with energy EV are trapped between them. Calculate the half-life 1/2 of a population
of such particles.
[7 marks]
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