91影视

In Example 4.2. neither${\text{触唯触}}^{\text{2}}$nor$\text{触唯触}$are given units鈥攐nly proportionalities are used. Here we verify that the results are unaffected. The actual values given in the example are particle detection rates, in particles/second, or${\text{s}}^{\text{-1}}$. For this quantity, let us use the symbol R. It is true that the particle detection rate and the probability density will be proportional, so we may write${\text{触唯触}}^{\text{2}}$= bR, where b is the proportionality constant. (b) What must be the units of b? (b) What is$\left|{\text{唯}}_{\text{T}}\right|$at the center detector (interference maximum) in terms of the example鈥檚 given detection rate and b? (c) What would be$\left|{\text{唯}}_{\text{1}}\right|\text{,}{\left|{\text{唯}}_{\text{1}}\right|}^{\text{2}}$, and the detection rate R at the center detector with one of the slits blocked?

Generally speaking, why is the wave nature of matter so counterintuitive?

Calculate the ratio of (a) energy to momentum for a photon, (b) kinetic energy to momentum for a relativistic massive object of speed u, and (e) total energy to momentum for a relativistic massive object. (d) There is a qualitative difference between the ratio in part (a) and the other two. What is it? (e) What are the ratios of kinetic and total energy to momentum for an extremelyrelativistic massive object, for which$\text{u}鈮�c\text{?}\underset{未x鈫�0}{lim}$ What about the qualitative difference now?

Classically and nonrelativistically, we say that the energy$\mathbf{\text{E}}$of a massive free particle is just its kinetic energy. (a) With this assumption, show that the classical particle velocity${\mathbf{\text{v}}}_{\mathbf{\text{particle}}}$is$\mathbf{\text{2E}}\mathbf{/}\mathbf{\text{p}}$. (b) Show that this velocity and that of the matter wave differ by a factor of 2. (c) In reality, a massive object also has internal energy, no matter how slowly it moves, and its total energy$\mathbf{\text{E}}$is$\mathit{纬}{\mathbf{\text{mc}}}^{\mathbf{\text{2}}}$, where_{$\mathit{纬}\mathbf{=}\mathbf{\text{1}}\mathbf{/}\sqrt{\mathbf{\text{1-}}{\mathbf{(}{\mathbf{\text{v}}}_{\mathbf{\text{particle}}}\mathbf{/}\mathbf{\text{c}}\mathbf{)}}^{\mathbf{\text{2}}}}$.}Show that${\mathbf{\text{v}}}_{\mathbf{\text{particle}}}$is${\mathbf{\text{pc}}}^{\mathbf{\text{2}}}\mathbf{/}\mathbf{\text{E}}$and that_{${\mathbf{\text{v}}}_{\mathbf{\text{wave}}}$}is${\mathbf{\text{c}}}^{\mathbf{\text{2}}}\mathbf{/}{\mathbf{\text{v}}}_{\mathbf{\text{particle}}}$Is there anything wrong with it_{${\mathbf{\text{v}}}_{\mathbf{\text{wave}}}$}? (The issue is discussed further in Chapter 6.)

Nonrelativistically, the energy$\mathit{E}$of a free massive particle is just kinetic energy, and its momentumis. of course,$\mathbf{\text{mv}}$. Combining these with fundamental relationships (4-4) and (4-5), derive a formula relating (a) particle momentumto matter-wave frequency fand (b) particle energy$\mathit{E}$to the wavelength$\mathbf{\text{位}}$of a matter wave.

In Section 4.3, it is shown that$\mathit{唯}{\mathbf{\text{(x,t)=Ae}}}^{\mathbf{\text{颈(办虫-蝇}}\mathbf{t}\mathbf{\text{)}}}$satisfies the free-particle Schr枚dinger equation for all$x$and$t$, provided only that the constants are related by${\mathbf{\text{(}}\mathbf{鈩�}\mathbf{\text{k)}}}^{\mathbf{\text{2}}}\mathbf{/}\mathbf{\text{2m}}\mathbf{=}\mathit{鈩�}\mathit{蝇}$. Show that when the function,$\mathit{唯}\mathbf{\text{(x,t)=Acos(kx}}\mathbf{鈭�}\mathbf{\text{0x)}}$is plugged into the Schrodinger equation, the result cannot possibly hold for all values of x and t, no matter how the constants may be related.

Because we have found no way to formulate quantum mechanics based on a single real wave function, we have a choice to make. In Section 4.3,it is said that our choice of using complex numbers is a conventional one. Show that the free-particle Schrodinger equation (4.8) is equivalent to two real equations involving two real functions, as follows:

_{$\mathbf{-}\frac{{\mathbf{鈩�}}^{\mathbf{2}}{\mathbf{位}}^{\mathbf{2}}{\mathbf{唯}}_{\mathbf{1}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{鈭�}\mathbf{\text{m}}}\mathbf{=}\mathit{鈩�}\frac{\mathbf{鈭�}{\mathbf{唯}}_{\mathbf{2}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{鈭�}\mathbf{\text{t}}}$}and

$\mathbf{-}\frac{{\mathbf{鈩�}}^{\mathbf{2}}{\mathbf{位}}^{\mathbf{2}}{\mathbf{唯}}_{\mathbf{2}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{鈭�}\mathbf{\text{m}}}\mathbf{=}\mathit{鈩�}\frac{\mathbf{鈭�}{\mathbf{唯}}_{\mathbf{1}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}}{\mathbf{鈭�}\mathbf{\text{t}}}$

where $\mathit{唯}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}$is by definition ${\mathit{唯}}_{\mathbf{1}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}\mathbf{+}\mathbf{\text{i}}{\mathit{唯}}_{\mathbf{2}}\mathbf{\left(}\mathbf{\text{x,t}}\mathbf{\right)}$. How is the complex approach chosen in Section4.3more convenient than the alternative posed here?

In Section 4.3, we claim that in analyzing electromagnetic waves, we could handle the fieldsandtogether with complex numbers. Show that if we define an "electromagnetic field"$\mathbf{\text{G}}\mathbf{鈮�}\mathbf{\text{E+icB}}$, then the two of Maxwell's equations that link$\mathit{E}$and$\mathit{B}$.$\mathbf{\text{(4-6c)}}$ and$\mathbf{\text{(4-6d)}}$ , become just one:

$\mathbf{鈭�}\mathbf{\text{G}}\mathbf{鈰�}\mathbf{\text{dl}}\mathbf{=}\frac{\mathbf{\text{i}}}{\mathbf{\text{c}}}\frac{\mathbf{鈭�}}{\mathbf{鈭�}\mathbf{\text{t}}}鈭�\text{G}\mathbf{鈰�}\mathbf{\text{dA}}$

Electromagnetic waves would have to obey this complex equation. Does this change of approach make $\mathit{E}$and$\mathit{B}$/or complex? (Remember how a complex number is defined.)

An electron moves along the x-axiswith a well-defined momentum of$\mathbf{5}\mathbf{脳}{\mathbf{10}}^{\mathbf{-}\mathbf{25}}\mathit{k}\mathit{g}\mathbf{.}\raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$. Write an expression describing the matter wave associated with this electron. Include numerical values where appropriate.

A free particle is represented by the plane wave function$\mathit{蠄}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}\mathit{A}\mathit{e}\mathit{x}\mathit{p}\mathbf{\left[}\mathbf{i}\left(1.58脳{10}^{12}x-7.91脳{10}^{16}t\right)\mathbf{\right]}$where SIunits are understood. What are the particle鈥檚 momentum, Kinetic energy, and mass? (Note: In nonrelativistic quantum mechanics, we ignore mass/internal energy, so the frequency is related to kinetic energy alone.)