91影视

A hydrogen atom starts out in the following linear combination of the stationary states n=2, l=1, m=1 and n=2, l=1, m=-1.

$\mathbf{蠄}\left(r,0\right)\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\left({蠄}_{211}+{蠄}_{21-1}\right)$

(a) Construct$\mathbf{蠄}\left(r,t\right)$Simplify it as much as you can.

(b) Find the expectation value of the potential energy,$<V>$. (Does it depend on t?) Give both the formula and the actual number, in electron volts.

(a) Prove that for a particle in a potential V(r)the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque:

$\frac{\mathbf{d}}{\mathbf{dt}}<L>\mathbf{=}<N>$

Where,

$\mathbf{N}\mathbf{=}\mathbf{r}\mathbf{脳}\left(\overline{V}V\right)$

(This is the rotational analog to Ehrenfest's theorem.)

(b) Show that $\mathbf{d}<L>\mathbf{/}\mathbf{dt}\mathbf{=}\mathbf{0}$for any spherically symmetric potential. (This is one form of the quantum statement of conservation of angular momentum.)

(a) Find the eigenvalues and eigenspinors of S_y .

(b) If you measured S_yon a particle in the general state X(Equation 4.139), what values might you get, and what is the probability of each? Check that the probabilities add up to 1 . Note: a and b need not be real!

(c) If you measured${\mathbf{S}}_{\mathbf{y}}^{\mathbf{2}}$ , what values might you get, and with what probabilities?

Work out the radial wave functions ${\mathit{R}}_{\mathbf{30}}\mathbf{,}{\mathit{R}}_{\mathbf{31}}$,and${\mathit{R}}_{\mathbf{32}}$using the recursion formula. Don鈥檛 bother to normalize them.

(a) Normalize${\mathit{R}}_{\mathbf{20}}$ (Equation 4.82), and construct the function${\mathit{蠄}}_{\mathbf{200}}$.

(b) Normalize${\mathit{R}}_{\mathbf{21}}$(Equation 4.83), and construct the function.

(a) Using Equation 4.88, work out the first four Laguerre polynomials.

(b) Using Equations 4.86, 4.87, and 4.88, find $\mathit{v}\left(蚁\right)$, for the case $\mathit{n}\mathbf{=}\mathbf{5}\mathbf{,}\mathit{I}\mathbf{=}\mathbf{2}$.

(c) Find $\mathit{v}\left(蚁\right)$again (for the case role="math" localid="1658315521558" $\mathbf{n}\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{I}\mathbf{=}\mathbf{2}$), but this time get it from the recursion formula (Equation 4.76).

$$\begin{gathered} L_q\left(x\right)=\frac{e^q}{q!}\left(\frac{d}{dx}{)}^q\right(e^{-x}-x^9)(4.88\left) \\ v\right(蚁)={L_{n-}^{2l+1}}_{l-1}(4.86\left) \\ {L}_q^p\right(x)鈮�(-1{)}^p{\left(\frac{d}{dx}\right)}^{蚁}{L}_{p+q}\left(x\right)(4.87) \\ {c}_{j+1}=\frac{2(j+l+1-n)}{(j+1)(j+2l+2)}{c}_j(4.76) \end{gathered}$$

(a) Find鈱﹔鈱猘nd鈱﹔虏鈱猣or an electron in the ground state of hydrogen. Express your answers in terms of the Bohr radius.

(b) Find鈱﹛鈱猘nd $\left(x^2\right)$for an electron in the ground state of hydrogen.

Hint: This requires no new integration鈥攏ote that ${\mathit{r}}^{\mathbf{2}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}{\mathit{z}}^{\mathbf{2}}$,and exploit the symmetry of the ground state.

(c) Find鈱﹛虏鈱猧n the state $\mathit{n}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{l}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{m}\mathbf{=}\mathbf{1}$. Hint: this state is not symmetrical in x, y, z. Use$\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{胃}\mathbf{}\mathbf{}\mathit{c}\mathit{o}\mathit{s}$$\mathit{蟺}\mathit{x}\mathbf{=}\mathit{r}\mathit{s}\mathit{i}\mathit{n}\mathit{胃}\mathit{c}\mathit{o}\mathit{s}\mathit{蠒}$

What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr.

A hydrogenic atom consists of a single electron orbiting a nucleus with Z protons. ($\mathbf{Z}\mathbf{=}\mathbf{1}$ would be hydrogen itself,$\mathit{Z}\mathbf{=}\mathbf{2}$is ionized helium ,$\mathit{Z}\mathbf{=}\mathbf{3}$is doubly ionized lithium, and so on.) Determine the Bohr energies $\mathit{E}\mathit{n}\mathbf{\left(}\mathit{Z}\mathbf{\right)}$, the binding energy${\mathit{E}}_{\mathbf{1}}\mathbf{\left(}\mathit{Z}\mathbf{\right)}$, the Bohr radius$\mathit{a}\mathbf{\left(}\mathit{Z}\mathbf{\right)}$, and the Rydberg constant R$\mathbf{\left(}\mathit{Z}\mathbf{\right)}$for a hydrogenic atom. (Express your answers as appropriate multiples of the hydrogen values.) Where in the electromagnetic spectrum would the Lyman series fall, for $\mathit{Z}\mathbf{=}\mathbf{2}$and $\mathit{Z}\mathbf{=}\mathbf{3}$? Hint: There鈥檚 nothing much to calculate here鈥� in the potential (Equation 4.52) $\mathit{Z}{\mathit{e}}^{\mathbf{2}}$, so all you have to do is make the same substitution in all the final results.

$\mathit{V}\mathbf{\left(}\mathit{r}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{{\mathbf{e}}^{\mathbf{2}}}{\mathbf{4}\mathbf{蟺}\stackrel{\mathbf{藱}}{{\mathbf{o}}_{\mathbf{0}}}}\frac{\mathbf{1}}{\mathbf{r}}$ (4.52).

Consider the earth鈥搒un system as a gravitational analog to the hydrogen atom.

(a) What is the potential energy function (replacing Equation 4.52)? (Let be the mass of the earth, and M the mass of the sun.)

$\mathit{V}\left(r\right)\mathbf{=}\mathbf{-}\frac{{\mathbf{e}}^{\mathbf{2}}}{\mathbf{4}\mathbf{蟺}{\displaystyle \stackrel{\mathbf{,}}{{\mathbf{0}}_{\mathbf{0}}}}}\frac{\mathbf{1}}{\mathbf{r}}$

(b) What is the 鈥淏ohr radius,鈥�${\mathit{a}}_{\mathbf{g}}$,for this system? Work out the actual number.

(c) Write down the gravitational 鈥淏ohr formula,鈥� and, by equating ${\mathbf{E}}_{\mathbf{n}}$to the classical energy of a planet in a circular orbit of radius ${\mathit{r}}_{\mathbf{0}}$, show that $\mathit{n}\mathbf{=}\sqrt{{\mathbf{r}}_{\mathbf{0}}\mathbf{/}{\mathbf{a}}_{\mathbf{g}}}$.From this, estimate the quantum number n of the earth.

(d) Suppose the earth made a transition to the next lower level$\left(n-1\right)$ . How much energy (in Joules) would be released? What would the wavelength of the emitted photon (or, more likely, gravitation) be? (Express your answer in light years-is the remarkable answer a coincidence?).