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Suppose that the electron in Fig. having a total energy E of 5.1 eV, approaches a barrier of height U_(b)=6.8eV and thickness L = 750 pm. (a) What is the approximate probability that the electron will be transmitted through the barrier, to appear (and be detectable) on the other side of the barrier? (b) What is the approximate probability that a proton with the same total energy of 5.1 eV will be transmitted through the barrier, to appear (and be detectable) on the other side of the barrier? |
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Answer» Solution :KEY IDEA The probability we seek is the transmission coefficient T as given by Eq. 37-22 `(T~~e^(-2bL))`, where `b=sqrt((8pi^(2)m(U_(b)-E))/(h^(2)))` Calculations : The numberator of the fraction under the square - root sign is `(8pi)(9.11xx106(-31)KG)(6.8eV-5.1eV)XX(1.60xx106(-19)J//eV)=1.965xx10^(-47)J.kg`. Thus, `b=sqrt((1.956xx106(-47)J.kg)/((6.63xx10^(-34)J.s)^(2)))=6.67xx10^(9)m^(-1)`. The (dimensionless) quantity 2bL is then `2bL=(2)(6.67xx10^(9)m^(-1))(750xx10^(-12)m)=10.0` and, from Eq. 37-22, the transmission coefficient is `T~~e^(-2bL)=e^(-100)=45xx10^(-6)` . Thus, of every million electrons that striks the barier, about 45 will tunnel through it, each appearing on the other side with its original total energy of 5.1 eV. (The transmission through the barrier does not alter in electron.s energy or any other property.) Reasoning : The transmission coefficient T (and thus the probability of transmission) depends on the mass of the particle. Indeed, because mass m is one of the factors in the exponent of e in the equation for T, the probability of transmission is very sensitive to the mass of the particle. This time, the mass is that of a proton `(1.67xx10^(-27)kg)`. which significantly greater than that of the electron in (a). By substituting the proton.s mass for the mass in (a) and then continuing as we did there, we find that `T~~10^(-186)`. Thus, although the probability that the proton will be transmitted is not exactly ZERO, it is barely more than zero. For even more massive particles with the same total energy of 5.1 eV, the probability of transmission is EXPONENTIALLY lower. |
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