- #1

docnet

- 587

- 242

- Homework Statement:
- psb

- Relevant Equations:
- psb

The equation $$\frac{\hbar^2}{2m}\frac{d^2u}{dr^2}-\frac{Ze^2}{r}u=Eu$$ gives the schrodinger equation for the spherically symmetric functions ##u=r\psi## for a hydrogen-like atom.

In this equation, substitute an assumed solution of the form ##u(r)=(Ar+Br^2)e^{-br}## and hence find the values of ##b## and the ratio ##B/A## for which this form of solution satisfies the equation. Verify that it corresponds to the second energy level, with ##E=Z^2/4## times the groud-state energy of hydrogen, and with ##B/A=-Z/2a_0## where ##a_0## is the Bohr radius for hydrogen. What is the value of the coefficient ##b## in terems of ##a_0##?

effort:

given

$$E_1=\frac{mZ^2e^4}{2\hbar^2}$$

and

$$a_0=\frac{4\pi\hbar^2}{Zme^2}\Rightarrow B/A= \frac{-2Z^2\pi\hbar^2}{me^2}$$

$$\frac{\hbar^2}{2m}[(A+2Br)e^{-br}-b(Ar+Br^2)e^{-br}]-\frac{Ze^2}{r}(Ar+Br^2)e^{-br}=\frac{mZ^4e^4}{8\hbar^2}(Ar+Br^2)e^{-br}$$

$$[(\frac{\hbar^2}{2mr}-\frac{b\hbar^2}{2m}-\frac{Ze^2}{r})Ar+(\frac{\hbar^2}{2mr}-\frac{b\hbar^2}{2m}-\frac{Ze^2}{r})Br^2]e^{-br}$$

$$(\frac{\hbar^2}{2mr}-\frac{b\hbar^2}{2m}-\frac{Ze^2}{r})(Ar+Br^2)e^{-br}$$

$$b=\frac{-m^2rZ^2e^4+h^4-2mZe^2h^2}{rh^4}$$

how do you proceed?

The book says the answer is $$b=\frac{Z}{2a_0}=\frac{2Z^2\pi\hbar^2}{me^2}$$

many thanks edited with progress

In this equation, substitute an assumed solution of the form ##u(r)=(Ar+Br^2)e^{-br}## and hence find the values of ##b## and the ratio ##B/A## for which this form of solution satisfies the equation. Verify that it corresponds to the second energy level, with ##E=Z^2/4## times the groud-state energy of hydrogen, and with ##B/A=-Z/2a_0## where ##a_0## is the Bohr radius for hydrogen. What is the value of the coefficient ##b## in terems of ##a_0##?

effort:

given

$$E_1=\frac{mZ^2e^4}{2\hbar^2}$$

and

$$a_0=\frac{4\pi\hbar^2}{Zme^2}\Rightarrow B/A= \frac{-2Z^2\pi\hbar^2}{me^2}$$

$$\frac{\hbar^2}{2m}[(A+2Br)e^{-br}-b(Ar+Br^2)e^{-br}]-\frac{Ze^2}{r}(Ar+Br^2)e^{-br}=\frac{mZ^4e^4}{8\hbar^2}(Ar+Br^2)e^{-br}$$

$$[(\frac{\hbar^2}{2mr}-\frac{b\hbar^2}{2m}-\frac{Ze^2}{r})Ar+(\frac{\hbar^2}{2mr}-\frac{b\hbar^2}{2m}-\frac{Ze^2}{r})Br^2]e^{-br}$$

$$(\frac{\hbar^2}{2mr}-\frac{b\hbar^2}{2m}-\frac{Ze^2}{r})(Ar+Br^2)e^{-br}$$

$$b=\frac{-m^2rZ^2e^4+h^4-2mZe^2h^2}{rh^4}$$

how do you proceed?

The book says the answer is $$b=\frac{Z}{2a_0}=\frac{2Z^2\pi\hbar^2}{me^2}$$

many thanks edited with progress

Last edited: