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Chapter 38: Q15Q (page 1181)

The table gives relative values for three situations for the barrier tunneling experiment of Figs. 38-16 and 38-17. Rank the situations according to the probability of the electron tunneling through the barrier, greatest first.

Electron Energy

Barrier Height

Barrier Thickness

(a)

E

5E

L

(b)

E

17E

L/2

(c)

E

2E

2L

Short Answer

Expert verified

The rank is Ta=Tb=Tc

Step by step solution

01

Describe the expression for probability of electron

The probability that is given by the transmission coefficient T is as follows.

T=e-2bL

where, b=8π2m(U-E)h2, U is barrier of height, and L is thickness.

According to these equations, the transmission coefficient decreases as the potential height increase, and also decreases as the energy of the incident beam decreases, it can be observed that, it depends on the value of U-E, as the value of U-Eincreases the transmission coefficient decrease. Also the transmission coefficient depends on width of the barrier, so,T∝e-LU-E

02

Rank the the probability of the electron tunneling through the barrier:

a) The transmission coefficient for the case (a) is given by,

Ta∝e-L4E∝e-2LE

b) The transmission coefficient for the case (b) is given by,

Tb∝e-L216E∝e-2LE

c) The transmission coefficient for the case (c) is given by,

Tc∝e-2L2E-E∝e-2LE

Therefore, all the transmission coefficients are equal.

Hence, the rank is Ta=Tb=Tc.

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Most popular questions from this chapter

Question: Figure 38-13 shows a case in which the momentum component

pxof a particle is fixed so that ∆px=0 ; then, from Heisenberg’s uncertainty principle (Eq. 38-28), the position x of the particle is completely unknown. From the same principle it follows that the opposite is also true; that is, if the position of a particle is exactly known (∆x=0), the uncertainty in its momentum is infinite.Consider an intermediate case, in which the position of aparticle is measured, not to infinite precision, but to within a distanceof λ2π, where λis the particle’s de Broglie wavelength.Show that the uncertainty in the (simultaneously measured) momentumcomponent is then equal to the component itself; that is,∆px=p. Under these circumstances, would a measured momentumof zero surprise you? What about a measured momentum of 0.5p? Of 0.2p? Of 12p?

Find the maximum kinetic energy of electrons ejected from a certain material if the material’s work function is 2.3 eV and the frequency of the incident radiation is 3.0×1015Hz.

Electrons accelerated to an energy of 50 GeVhave a de Broglie wavelength λsmall enough for them to probe the structure within a target nucleus by scattering from the structure. Assume that the energy is so large that the extreme relativistic relation p=Ecbetween momentum magnitude pand energy Eapplies. (In this extreme situation, the kinetic energy of an electron is much greater than its rest energy.)

(a) What is λ?

(b) If the target nucleus has radius R=5.0 fm, what is the ratio Rλ?

Light of wavelength 2.40 pm is directed onto a target containing free electrons. (a) Find the wavelength of light scattered at 30.00 from the incident direction. (b) Do the same for a scattering angle of 1200?

A nonrelativistic particle is moving three times as fast as an electron. The ratio of the de Broglie wavelength of the particle to that of the electron is 1.813×10−4. By calculating its mass, identify the particle.
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