Question

The function ψ2px-1(ψ2,1,1+ψ2,1-1) describes an electron in the 2px state of a hydrogen-like atom (with unspecified spin). Functions ψη..my are normalized egenfuntions of the energy operator (A), the square of angular momentum operator (12), and the z-component of angular momentum operator (Lz), that is 4. E1 a) Show that the function ψ2px is an eigen function of both the energy operator and the square of angular momentum operator. Find the corresponding eigenvalues. b) Determine the expected value and the uncertainty of energy in the 2p state. c) Determine the expected value of the z-component of the angular momentum.

Answer 1

$\psi _{2p_x } = \frac{1}{\sqrt{2}} (\psi _{2,1,1} + \psi_{2,1,-1})$

$\hat{H}\psi _{n,l,m } = \frac{E_1}{n^2} \psi _{n,l,m } \\ \hat{H}\psi _{2,1,1 } = \frac{E_1}{2^2} \psi _{2,1,1 } \\ \hat{H}\psi _{2,l,-1 } = \frac{E_1}{2^2} \psi _{2,1,-1 } \\ \hat{H}\psi _{2p_x } = \hat{H} \left [ \frac{1}{\sqrt{2}}\left ( \psi _{2,1,1} + \psi _{2,1,-1}\right ) \right ]$

$=\frac{1}{\sqrt{2}} \left [ \left ( \hat{H}\psi _{2,1,1} + \hat{H}\psi _{2,1,-1}\right ) \right ] \\ = \frac{1}{\sqrt{2}} \left [ \frac{E_1}{2^2} \left ( \psi _{2,1,1} + \psi _{2,1,-1}\right )\right ] \\ =\frac{E_1}{4} \left [ \frac{1}{\sqrt{2}} \left ( \psi _{2,1,1} + \psi _{2,1,-1}\right )\right ] \\ = \frac{E_1}{4}\psi_ {2p_x}$

eigen values is E₁ /4

$\hat{L^2}\psi_{n,l,m} = l(l+1)\not{h^2}\psi_{n,l,m}$

$\hat{L^2}\psi _{2,1,1 } = 1(1+1) \not{h^2}\psi _{2,1,1 } \\ \hat{L^2}\psi _{2,l,-1 } = 1(1+1) \not{h^2} \psi _{2,1,-1 } \\ \hat{L^2}\psi _{2p_x } = \hat{L^2} \left [\left ( \psi _{2,1,1} + \psi _{2,1,-1}\right ) \right ] \\ = 2\not{h^2} \left ( \psi _{2,1,1} + \psi _{2,1,-1}\right ) \\ = 2\not{h^2} \psi _{2p_x}$

eigen value is $2\not{h^2} \\ \psi _{2p_x} is\; an \;eigen \; function \; of \;both \;\hat{H} and \hat{L^2}$

b) [H] energy is a sharp observable as $\psi _{2p_x}$ is an eigen function of the operator. The expected value is the energy is eigen value E₁ /4 nd the uncertainty is 0

c)  

$\hat{L_z} \psi _{n,l,m }= m\not{h} \psi _{n,l,m } \\$

the eigen value is $m_l\not{h}$ is is sharp observable. Uncertainty is 0, expected vlue of z-component is the eigen value