PLEASE SHOW ALL WORK a) Consider a particle incident left to right on a barrier

Question

PLEASE SHOW ALL WORK

a) Consider a particle incident left to right on a barrier potential, as in the figure. What is the particle's wave- V(x) function if its energy is E Vo? And if E Vo? You do not need to complete the full calculation, but explain in detail how you would get to the result and draw the Particle waveform, indicating clearly the difference between re- gions I, II and III. b) "Quantum tunneling" or "barrier penetration" is not Region I Region II Region III an experience of everyday life. A sprinter of mass 70 kg running at 5 m/s does not have enough kinetic energy to leap a wall of height 5 meters, even if all of that kinetic energy could be directed into an upward leap. If the wall is 0.2 meters thick, estimate the probability of the sprinter being able to "quantum-tunnel" through it, rather than leaping it.

Explanation / Answer

a) For the case E>V0 the wave function is a simple wave. Its general expression is

In region I)

Psi(x) = A1*exp(i*k1*x) + A2*exp(-i*k1*x)

(where k is the wave number and needs to be REAL)

First term A1*exp(i*k1*x) is the transmitted part, second term A2*exp(-i*k2*x)) is the reflected term

k1 = 2*pi/lambda = p/h_bar = sqrt(2*m*E)/h_bar

In region II) we have

Psi(x) = B1*exp(i*k2*x)+B2*exp(-i*k2*x)

k2 = 2*pi/lambda =p/h_bar =sqrt(2*m*(E-V0))/h_bar

In region III)

Psi(x) = C1*exp(i*k3*x) +C2*exp(-ik3*x)

k3 = 2*pi/lambda = p/h_bar = sqrt(2*m*E)/h_bar

The coefficients need to satisfy the continuity conditions for Psi(x) and dPsi/dx at x =0 and x=L

That is:

A1+A2 = B1+B2

B1*exp(i*k2*L) +B2*exp(..) =C1*exp(i*k3*L) +C2*exp(..)

ik1A1 -ik2A2 =ik2B1 -ik2B2

ik2B1*exp(ik2L) -ik2B2exp(-ik2L) = ik3C1*exp(ik3L) -ik3C2*exp(...)

b) in this case (E < V0) only the expression in region II is modified from above. In region I and III) there will be the same expressions for Psi as above.

In region II we have) (V0 > E) an exponential real function

Psi(x) = B1*exp(k2*x) +Bx*exp(-k2*x)

where k2 is the wave number REAL

k2 = sqrt(2*m*(V0-E))/ h_bar

The same onditions for Psi(x) and dPsi/dx at x=0 and x =L apply

Question B) with sprinter

Suppose one has only the transmitted wave

In region I) we have a wave

Psi(x) = A*exp(i*k1*x)

In region II) an exponentially decreasing function

Psi(x) = B*exp(-k2*x)

In region III) we have again the direct transmitted wave

Psi(x) = C*exp(i*k3*x)

Continuity of psi(x) at x= 0 and x =L

A =B

B*exp(-k2*L) =C*exp(i*k3*L)

C = A*exp(-k2*L)/exp(i*k3*L) = A*exp(-k2*L)*exp(-i*k3*L)

The ratio of the amplitudes of wavefunction in regions III and I) is:

Psi_III/Psi_I = C*exp(ik3x)/A*exp(i*k1*x)

The probability of tunneling (coefficient of transmission) is just the square of the modulus of the ratio:

T = | Psi_III/Psi_I |^2 = |C/A|^2 = exp(-2*k2*L)

V0 = m*g*h =70*9.81*5 =3433.5 J

E =mv^2/2 =70*5^2/2 = 875 J

L =0.2 m

k2 = sqrt(2*m*(V0-E))/h_bar =sqrt(2*70*(3433.5-875)) /10^-34 =6*10^36

T =exp(-2*k2*L) = exp(-2*6*10^36*0.2) =.....

The windows scientific calculator can not compute such a small value.

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