Chapter 7: Problem 68
Roughly, how does the size of a triply ionized beryllium ion compare with hydrogen?
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A spherical infinite well has potential energy $$ U(r)=\left\\{\begin{array}{ll} 0 & ra \end{array}\right. $$ Since this is a central force, we may use the Schrödinger equation in the form \((7-30)-\) - that is, just before the specific hydrogen atom potential energy is inserted. Show that the following is a solution $$ R(r)=\frac{A \sin b r}{r} $$ Now apply the appropriate boundary conditions, and in so doing, find the allowed angular momenta and energies for solutions of this form.
Imagine two classical charges of \(-q\), each bound to a central charge of \(+4 .\) One \(-q\) charge is in a circular orbit of radius \(R\) about its \(+q\) charge. The other oscillates in an extreme ellipse, essentially a straight line from its \(+q\) charge out to a maximum distance \(r_{\max }\) The two orbits have the same energy. (a) Show that \(r_{\max }=2 R .\) (b) Considering the time spent at each orbit radius, in which orbit is the \(-q\) charge farther from its \(+q\) charge on average?
An electron is confined to a cubic \(3 \mathrm{D}\) infinite well \(1 \mathrm{~nm}\) on a side. (a) Whut ure the three lowest different energies possible? (b) To how many different states do these three energies correspond?
The 20 Infinile Wellt In imis dimensiona, the Schrodinger equation is
$$
\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{a^{2}}{a y^{2}}\right)
\mapsto(x, y)=-\frac{2 m(E-U)}{A^{2}} \psi(x, y)
$$
(a) Given that \(t\) is a constant. separate variables by trying a solution of
the form \(\phi(x, y)=f(x) g(y)\). then diviting by \(f(n) g(v)\). Call the
separution constants \(C\), and \(C_{y^{2}}\)
(b) For an inlinite well.
$$
U=\left\\{\begin{array}{cc}
0 & 0
For an elecTron in the \(\left(n, C, m_{e}\right)=(2,0,0)\) state in a hydrogen atom. (a) write the solution of the time-independent Schrödinger equation, and (b) verify explicitly that it is a solution with the expected angular momentum and eneRgy.
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