91影视

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?

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.

The uncertainty in the position of a baseball of mass$\mathbf{0}\mathbf{.}\mathbf{145}\mathbf{}\mathit{k}\mathit{g}\mathbf{}\mathit{i}\mathit{s}\mathbf{}\mathit{渭}\mathit{m}$.What is the minimum uncertainty in its speed?

To how small a region must an electron be confined for borderline relativistic speeds say$\mathbf{\text{0.05}}$to become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, which has an orbit radius near${\mathbf{\text{10}}}^{\mathbf{\text{-10}}}\mathbf{\text{m}}$? For a lead atom "inner-shell" electron of orbit radius${\mathbf{\text{10}}}^{\mathbf{\text{-12}}}\mathbf{\text{m}}$?

One of the cornerstones of quantum mechanics is that bound particles cannot be stationary-even at zero absolute temperature! A "bound" particle is one that is confined in some finite region of space. as is an atom in a solid. There is a nonzero lower limit on the kinetic energy of such a particle. Suppose minimum kinetic energy of width $\mathbf{L}$. Obtain an approximate formula for its minimum kinetic energy.

An electron in an atom can "jump down" from a higher energy level to a lower one, then to a lower one still. The energy the atom thus loses at each jump goes to a photon. Typically, an electron might occupy a level for a nanosecond. What uncertainty in the electron's energy does this imply?

A crack between two walls is $10\text{鈥塩尘}$wide. What is the angular width of the central diffraction maximum when

(a) an electron moving at $50\text{鈥尘/蝉}$passes through?

(b) A baseball of mass$0.145\text{鈥塳驳}$ and speed $50\text{鈥尘/蝉}$passes through?

(c) In each case, an uncertainty in momentum is introduced by the 鈥渆xperiment鈥� (i.e., passing through the slit). Specifically, what aspect of the momentum becomes uncertain, and how does this uncertainty compare with the initial momentum of each?

In Exercise 45, the case is made that the position uncertainty for a typical macroscopic object is generally so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. Here we gain same idea of how small an object would have鈾� to be before quantum mechanics might rear its head. The density of aluminum is $\mathbf{2}\mathbf{.7}\mathbf{脳}{\mathbf{10}}^{\mathbf{3}}\mathbf{\text{鈥�}}\mathbf{}\mathbf{\text{kg}}\mathbf{/}{\mathbf{\text{m}}}^{\mathbf{3}}$, is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty of$\mathbf{1}\mathbf{渭}\mathbf{\text{m}}$per decade. How small would it have to be for its position uncertainty to be at least as large as$\frac{\mathbf{1}}{\mathbf{10}}\mathit{\%}$of its radius?