91影视

Verify that (9.15) follows from (9.14). Hint: Use the formulas for $\mathit{t}\mathit{a}\mathit{n}\mathbf{(}\mathbf{伪}\mathbf{卤}\mathbf{尾}\mathbf{)}$, $\mathbf{tan}\mathbf{2}\mathit{伪}$, etc., to condense (9.14) and then change to polar coordinates. You may find

$\mathit{u}\mathbf{=}\frac{\mathbf{100}}{\mathbf{蟺}}\mathbf{arctan}\frac{\mathbf{sin}\mathbf{2}\mathbf{胃}}{{\mathbf{r}}^{\mathbf{2}}\mathbf{鈭�}\mathbf{cos}\mathbf{2}\mathbf{胃}}$

Show that if you use principal values of the arc tangent, this formula does not give the correct boundary conditions on the x-axis, whereas (9.15) does.

Show that the gravitational potential $\mathit{V}\mathbf{=}\mathbf{鈭�}\frac{\mathbf{G}\mathbf{m}}{\mathbf{r}}$satisfies Laplace's equation, that is, show that ${\mathbf{鈭�}}^{\mathbf{2}}\mathbf{\left(}\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{r}$}\right.\mathbf{\right)}\mathbf{=}\mathbf{0}$where${\mathit{r}}^{\mathbf{2}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathbf{,}\mathit{r}\mathbf{鈮�}\mathbf{0}$.

Use Problem 7.16 to find the characteristic vibration frequencies of sound in a spherical cavity.

Write the Schr枚dinger equation (3.22) if $\mathit{蠄}$is a function ofx, and $\mathit{V}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{蝇}}^{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}$ (this is a one-dimensional harmonic oscillator). Find the solutions ${\mathit{蠄}}_{\mathbf{n}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and the energy eigenvalues ${\mathit{E}}_{\mathbf{n}}$ . Hints: In Chapter 12, equation (22.1) and the first equation in (22.11), replace xby $\mathit{伪}\mathit{x}$where $\mathit{伪}\mathbf{=}\sqrt{\mathbf{m}\mathbf{蝇}\mathbf{/}\mathbf{鈩�}}$. (Don't forget appropriate factors of $\mathit{伪}$for the ${\mathit{x}}^{\mathbf{\text{'}}}$ 's in the denominators of $\mathit{D}\mathbf{=}\raisebox{1ex}{$\mathbf{d}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}\mathbf{x}$}\right.$and ${\mathit{蠄}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{=}\raisebox{1ex}{${\mathbf{d}}^{\mathbf{2}}\mathbf{蠄}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}{\mathbf{x}}^{\mathbf{2}}$}\right.$.) Compare your results for equation (22.1) with the Schr枚dinger equation you wrote above to see that they are identical if ${\mathit{E}}_{\mathbf{n}}\mathbf{=}\mathbf{(}\mathbf{n}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{)}\mathit{鈩�}\mathit{蝇}$. Write the solutions ${\mathit{蠄}}_{\mathbf{n}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$of the Schr枚dinger equation using Chapter 12, equations (22.11) and (22.12).

The surface temperature of a sphere of radius 1 is held at $\mathit{u}\mathbf{=}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{胃}\mathbf{+}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{3}}\mathit{胃}$. Find the interior temperature $\mathit{u}\left(r,胃,蠒\right)$.

Separate the Schr枚dinger equation (3.22) in rectangular coordinates in 3 dimensions assuming that $\mathit{V}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{蝇}}^{\mathbf{2}}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{)}$. (This is a 3-dimensional harmonic oscillator). Observe that each of the separated equations is of the form of the one-dimensional oscillator equation in Problem 20. Thus write the solutions ${\mathit{蠄}}_{\mathbf{n}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}$for the 3dimensional problem, where, find the energy eigenvalues ${\mathit{E}}_{\mathbf{n}}$and their degree of degeneracy (see Problem (6.7) and Chapter 15, Problem 4.21).

Find the interior temperature in a hemisphere if the curved surface is held at $\mathit{u}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{胃}$and the equatorial plane at $\mathit{u}\mathbf{=}\mathbf{1}$.

Find the energy eigenvalues and Eigen functions for the hydrogen atom. The potential energy is $\mathit{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}\mathbf{鈭�}{\mathit{e}}^{\mathbf{2}}\mathbf{/}\mathit{r}$ in Gaussian units, where is the charge of the electron and r is in spherical coordinates. Since V is a function of r only, you know from Problem 18 that the Eigen functions are R(r) times the spherical harmonics ${\mathit{Y}}_{\mathbf{l}}^{\mathbf{m}}\mathbf{(}\mathbf{胃}\mathbf{,}\mathbf{蠒}\mathbf{)}$, so you only have to find R(r). Substitute V(r) into the R equation in Problem 18 and make the following simplifications: Let $\mathit{x}\mathbf{=}\raisebox{1ex}{$\mathbf{2}\mathbf{r}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{伪}$}\right.\mathbf{,}\mathit{y}\mathbf{=}\mathit{r}\mathit{R}$; show that then

$\mathit{r}\mathbf{=}\raisebox{1ex}{$\mathbf{伪}\mathbf{x}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathit{R}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}}{\mathbf{伪}\mathbf{x}}\mathit{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{r}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{伪}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{r}}\mathbf{\left(}{\mathbf{r}}^{\mathbf{2}}\frac{\mathbf{d}\mathbf{R}}{\mathbf{d}\mathbf{r}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}}{\mathbf{伪}}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}$. Let ${\mathit{伪}}^{\mathbf{2}}\mathbf{=}\mathbf{鈭�}\mathbf{2}\mathit{M}\mathit{E}\mathbf{/}{\mathit{鈩�}}^{\mathbf{2}}$(note that for a bound state, E is negative, so ${\mathit{伪}}^{\mathbf{2}}$is positive) and $\mathit{位}\mathbf{=}\mathit{M}{\mathit{e}}^{\mathbf{2}}\mathit{伪}\mathbf{/}{\mathit{鈩�}}^{\mathbf{2}}$, to get the first equation in Problem 22.26 of Chapter 12. Do this problem to find y(x) , and the result that $\mathit{位}$is an integer, say n .[Caution: not the same n as in equation (22.26)]. Hence find the possible values of $\mathit{伪}$(these are the radii of the Bohr orbits), and the energy eigenvalues. You should have found $\mathit{伪}$ proportional to n; let $\mathit{伪}\mathbf{=}\mathit{n}\mathit{a}$, where ais the value of $\mathit{伪}$when n = 1, that is, the radius of the first Bohr orbit. Write the solutions R(r) by substituting back $\mathit{y}\mathbf{=}\mathit{r}\mathit{R}$, and $\mathit{x}\mathbf{=}\mathbf{2}\mathit{r}\mathbf{/}\mathbf{\left(}\mathbf{n}\mathbf{a}\mathbf{\right)}$, and find ${\mathit{E}}_{\mathbf{n}}$from$\mathit{伪}$.

Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants, etc.$\mathit{T}\mathbf{=}\mathit{A}\mathbf{,}\mathbf{鈥�}\mathbf{鈥�}\mathit{T}\mathbf{=}\mathit{B}$, on the four sides, and the rectangle covers the area $\mathbf{0}\mathbf{<}\mathit{x}\mathbf{<}\mathit{a}\mathbf{,}\mathbf{鈥�}\mathbf{0}\mathbf{<}\mathit{y}\mathbf{<}\mathit{b}$.

The Klein-Gordon equation is ${\mathbf{鈭�}}^{\mathbf{2}}\mathit{u}\mathbf{=}\left(1/V^2\right){\mathbf{鈭�}}^{\mathbf{2}}\mathit{u}\mathbf{/}\mathbf{鈭�}{\mathit{t}}^{\mathbf{2}}\mathbf{+}{\mathit{位}}^{\mathbf{2}}\mathit{u}$. This equation is of interest in quantum mechanics, but it also has a simpler application. It describes, for example, the vibration of a stretched string which is embedded in an elastic medium. Separate the one-dimensional Klein-Gordon equation and find the characteristic frequencies of such a string.