91影视

The half-life of $\left(t_{1/2}\right)$an excited state is the time it would take for half the atoms in a large sample to make a transition. Find the relation betweenrole="math" localid="1658300900358" ${\mathbf{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{}}\mathbf{and}\mathbf{}\mathbf{T}$(the 鈥渓ife time鈥� of the state).

Calculate the lifetime (in seconds) for each of the four n = 2 states of hydrogen. Hint: You鈥檒l need to evaluate matrix elements of the form $<{蠄}_{100}\left|x\right|{蠄}_{200}>,<{蠄}_{100}\left|y\right|{蠄}_{211}>$, and so on. Remember that role="math" localid="1658303993600" $\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathbf{蝉颈苍胃肠辞蝉}\mathbf{}\mathit{蠒}\mathbf{,}\mathit{y}\mathbf{=}\mathit{r}\mathbf{}\mathbf{蝉颈苍胃蝉颈苍}\mathbf{}\mathit{蠒}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{z}\mathbf{=}\mathit{r}\mathbf{}\mathbf{肠辞蝉胃}$. Most of these integrals are zero, so inspect them closely before you start calculating. Answer: $\mathbf{1}\mathbf{.}\mathbf{60}\mathbf{脳}{\mathbf{10}}^{\mathbf{-}\mathbf{9}}$seconds for all except role="math" localid="1658304185040" ${\mathbf{蠄}}_{\mathbf{200}}$, which is infinite.

Prove the commutation relation in Equation 9.74. Hint: First show that

$\mathbf{[}{\mathbf{L}}^{\mathbf{2}}\mathbf{,}\mathbf{z}\mathbf{]}\mathbf{=}\mathbf{2}\mathit{i}\mathit{h}\mathbf{(}\mathbf{x}{\mathbf{L}}_{\mathbf{y}}\mathbf{-}\mathbf{y}{\mathbf{L}}_{\mathbf{x}}\mathbf{-}\mathbf{i}\mathbf{h}\mathbf{z}\mathbf{)}$

Use this, and the fact that localid="1657963185161" $\mathbf{r}\mathbf{.}\mathbf{L}\mathbf{=}\mathbf{r}\mathbf{.}\mathbf{(}\mathbf{r}\mathbf{脳}\mathbf{p}\mathbf{)}\mathbf{=}\mathbf{0}$, to demonstrate that

$\mathbf{[}{\mathbf{L}}^{\mathbf{2}}\mathbf{,}\mathbf{[}{\mathbf{L}}^{\mathbf{2}}\mathbf{,}\mathbf{z}\mathbf{]}\mathbf{]}\mathbf{=}\mathbf{2}{\mathit{h}}^{\mathbf{2}}\mathbf{(}\mathbf{z}{\mathbf{L}}^{\mathbf{2}}\mathbf{+}{\mathbf{L}}^{\mathbf{2}}\mathbf{z}\mathbf{)}$

The generalization from z to r is trivial.

Close the 鈥渓oophole鈥� in Equation 9.78 by showing that if${\mathit{l}}^{\mathbf{\text{'}}}\mathbf{=}\mathit{l}\mathbf{=}\mathbf{0}$then$\mathbf{鉄�}{\mathbf{n}}^{\mathbf{\text{'}}}{\mathbf{l}}^{\mathbf{\text{'}}}{\mathbf{m}}^{\mathbf{\text{'}}}\mathbf{\left|}\mathbf{r}\mathbf{\right|}\mathbf{n}\mathbf{l}\mathbf{m}\mathbf{鉄�}\mathbf{=}\mathbf{0}$

An electron in the $\mathit{n}\mathbf{=}\mathbf{3}\mathbf{,}\mathit{l}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{m}\mathbf{=}\mathbf{0}$state of hydrogen decays by a sequence of (electric dipole) transitions to the ground state.

(a) What decay routes are open to it? Specify them in the following way:

$\mathbf{|}\mathbf{300}\mathbf{鉄�}\mathbf{鈫�}\mathbf{|}\mathit{n}\mathcal{l}\mathit{m}\mathbf{鉄�}\mathbf{鈫�}\mathbf{|}{\mathbf{n}}^{\mathbf{\text{'}}}{\mathbf{l}}^{\mathbf{\text{'}}}{\mathbf{m}}^{\mathbf{\text{'}}}\mathbf{鉄�}\mathbf{鈫�}\mathbf{鈰�}\mathbf{鈫�}\mathbf{|}\mathbf{100}\mathbf{鉄�}\mathbf{.}$

(b) If you had a bottle full of atoms in this state, what fraction of them would decay via each route?

(c) What is the lifetime of this state? Hint: Once it鈥檚 made the first transition, it鈥檚 no longer in the state |300\rangle鈭�300鉄�, so only the first step in each sequence is relevant in computing the lifetime.

Develop time-dependent perturbation theory for a multi-level system, starting with the generalization of Equations 9.1 and 9.2:

${\widehat{\mathbf{H}}}_{\mathbf{0}}{\mathit{蠄}}_{\mathbf{n}}\mathbf{=}{\mathit{E}}_{\mathbf{n}}{\mathit{蠄}}_{\mathbf{n}}\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathbf{鉄�}{\mathbf{蠄}}_{\mathbf{n}}\mathbf{鈭�}{\mathbf{蠄}}_{\mathbf{m}}\mathbf{鉄�}\mathbf{=}{\mathit{未}}_{\mathbf{n}\mathbf{m}}$ (9.79)

At time t = 0 we turn on a perturbation ${\mathit{H}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$so that the total Hamiltonian is

$\widehat{\mathbf{H}}\mathbf{=}{\widehat{\mathbf{H}}}_{\mathbf{0}}\mathbf{+}{\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$(9.80).

(a) Generalize Equation 9.6 to read

$\mathit{唯}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{c}}_{\mathbf{a}}\mathbf{\left(}\mathit{t}\mathbf{\right)}{\mathit{蠄}}_{\mathbf{a}}{\mathit{e}}^{\mathbf{鈭�}\mathbf{i}{\mathbf{E}}_{\mathbf{a}}\mathbf{t}\mathbf{/}\mathbf{鈩�}}\mathbf{+}{\mathit{c}}_{\mathbf{b}}\mathbf{\left(}\mathit{t}\mathbf{\right)}{\mathit{蠄}}_{\mathbf{b}}{\mathit{e}}^{\mathbf{鈭�}\mathbf{i}{\mathbf{E}}_{\mathbf{b}}\mathbf{t}\mathbf{/}\mathbf{鈩�}}$(9.81).

and show that

${\stackrel{\mathbf{藱}}{\mathbf{c}}}_{\mathbf{m}}\mathbf{=}\mathbf{鈭�}\frac{\mathbf{i}}{\mathbf{鈩�}}\underset{n}{鈭�}{c}_n{\mathit{H}}_{\mathbf{m}\mathbf{n}}^{\mathbf{\text{'}}}{\mathit{e}}^{\mathbf{i}\mathbf{(}{\mathbf{E}}_{\mathbf{m}}\mathbf{鈭�}{\mathbf{E}}_{\mathbf{n}}\mathbf{)}\mathbf{t}\mathbf{/}\mathbf{鈩�}}$ (9.82).

Where

${\mathit{H}}_{\mathbf{m}\mathbf{n}}^{\mathbf{\text{'}}}\mathbf{鈮�}{\mathit{蠄}}_{\mathbf{m}}\mathbf{\left|}{\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}\mathbf{\right|}{\mathit{蠄}}_{\mathbf{n}}$ (9.83).

(b) If the system starts out in the state ${蠄}_N$, show that (in first-order perturbation theory)

${\mathit{c}}_{\mathbf{N}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{鈮�}\mathbf{1}\mathbf{鈭�}\frac{\mathbf{i}}{\mathbf{鈩�}}{鈭�}_0^t{H}_{NN}^{\text{'}}\mathbf{\left(}{\mathbf{t}}^{\mathbf{\text{'}}}\mathbf{\right)}\mathit{d}{\mathit{t}}^{\mathbf{\text{'}}}$(9.84).

and

${\mathit{c}}_{\mathbf{m}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{鈮�}\mathbf{鈭�}\frac{\mathbf{i}}{\mathbf{鈩�}}{鈭�}_0^t{H}_{mN}^{\text{'}}\mathbf{\left(}{\mathbf{t}}^{\mathbf{\text{'}}}\mathbf{\right)}{\mathit{e}}^{\mathbf{i}\mathbf{(}{\mathbf{E}}_{\mathbf{m}}\mathbf{鈭�}{\mathbf{E}}_{\mathbf{N}}\mathbf{)}{\mathbf{t}}^{\mathbf{\text{'}}}\mathbf{/}\mathbf{鈩�}}\mathit{d}{\mathit{t}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathbf{(}\mathit{m}\mathbf{鈮�}\mathit{N}\mathbf{)}$(9.85).

(c) For example, suppose${\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}$is constant (except that it was turned on at t = 0 , and switched off again at some later time . Find the probability of transition from state N to state M $\mathbf{(}\mathit{M}\mathbf{鈮�}\mathit{N}\mathbf{)}\mathbf{,}$as a function of T. Answer:

$\mathbf{4}{\mathbf{\left|}{\mathbf{H}}_{\mathbf{M}\mathbf{N}}^{\mathbf{\text{'}}}\mathbf{\right|}}^{\mathbf{2}}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{[}\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{鈭�}{\mathbf{E}}_{\mathbf{M}}\mathbf{)}\mathbf{T}\mathbf{/}\mathbf{2}\mathbf{鈩�}\mathbf{]}}{{\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{鈭�}{\mathbf{E}}_{\mathbf{M}}\mathbf{)}}^{\mathbf{2}}}$ (9.86).

(d) Now suppose${\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}$is a sinusoidal function of time${\widehat{\mathbf{H}}}^{\mathbf{\text{'}}}\mathbf{=}\mathit{V}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{蝇}\mathit{t}\mathbf{\right)}$: Making the usual assumptions, show that transitions occur only to states with energy ${\mathit{E}}_{\mathbf{M}}\mathbf{=}{\mathit{E}}_{\mathbf{N}}\mathbf{卤}\mathit{鈩�}$, and the transition probability is

${\mathit{P}}_{\mathbf{N}\mathbf{鈫�}\mathbf{M}}\mathbf{=}{\mathbf{\left|}{\mathbf{V}}_{\mathbf{M}\mathbf{N}}\mathbf{\right|}}^{\mathbf{2}}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{[}\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{鈭�}{\mathbf{E}}_{\mathbf{M}}\mathbf{卤}\mathbf{鈩�}\mathbf{蝇}\mathbf{)}\mathbf{T}\mathbf{/}\mathbf{2}\mathbf{鈩�}\mathbf{]}}{{\mathbf{(}{\mathbf{E}}_{\mathbf{N}}\mathbf{鈭�}{\mathbf{E}}_{\mathbf{M}}\mathbf{卤}\mathbf{鈩�}\mathbf{蝇}\mathbf{)}}^{\mathbf{2}}}$ (9.87).

(e) Suppose a multi-level system is immersed in incoherent electromagnetic radiation. Using Section 9.2.3 as a guide, show that the transition rate for stimulated emission is given by the same formula (Equation 9.47) as for a two-level system.

${\mathit{R}}_{\mathbf{b}\mathbf{鈫�}\mathbf{a}}\mathbf{=}\frac{\mathbf{蟺}}{\mathbf{3}{\mathbf{系}}_{\mathbf{0}}{\mathbf{鈩�}}^{\mathbf{2}}}\mathbf{|}\mathit{鈩�}{\mathbf{|}}^{\mathbf{2}}\mathit{蚁}\mathbf{\left(}{\mathbf{蝇}}_{\mathbf{0}}\mathbf{\right)}\mathit{R}\mathit{b}$ (9.47).

A particle starts out (at time t=0 ) in the Nth state of the infinite square well. Now the 鈥渇loor鈥� of the well rises temporarily (maybe water leaks in, and then drains out again), so that the potential inside is uniform but time dependent:${\mathit{V}}_{\mathbf{0}}\left(t\right)\mathbf{,}\mathbf{}\mathbf{with}\mathbf{}{\mathit{V}}_{\mathbf{0}}\left(0\right)\mathbf{=}{\mathit{V}}_{\mathbf{0}}\left(T\right)\mathbf{=}\mathbf{0}$.

(a) Solve for the exact ${\mathit{c}}_{\mathbf{m}}\left(t\right)$, using Equation 11.116, and show that the wave function changes phase, but no transitions occur. Find the phase change, role="math" localid="1658378247097" $\mathbf{蠒}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, in terms of the function ${\mathit{V}}_{\mathbf{0}}\left(t\right)$

(b) Analyze the same problem in first-order perturbation theory, and compare your answers. Compare your answers.
Comment: The same result holds whenever the perturbation simply adds a constant (constant in x, that is, not in to the potential; it has nothing to do with the infinite square well, as such. Compare Problem 1.8.

A particle of mass m is initially in the ground state of the (one-dimensional) infinite square well. At time t = 0 a 鈥渂rick鈥� is dropped into the well, so that the potential becomes

$\mathit{V}\left(x\right)\mathbf{=}\{\begin{array}{ccc}{V}_0,& 0鈮�x鈮�a/2& \\ 0,& a/2<x鈮�a;& \\ 鈭�,& \mathrm{otherwise}& \end{array}$

where ${\mathit{V}}_{\mathbf{0}}\mathbf{鈮�}{\mathit{E}}_{\mathbf{1}}$After a time T, the brick is removed, and the energy of the particle is measured. Find the probability (in first-order perturbation theory) that the result is now${\mathit{E}}_{\mathbf{2}}$ .

We have encountered stimulated emission, (stimulated) absorption, and spontaneous emission. How come there is no such thing as spontaneous absorption?