Question

( -hbar ), an ( L^{2}=2 hbar ). Consider three wavefunctions [ left{egin{aligned} psi_{+1} & =frac{-1}{A}(x+i y) Rleft(x^{2}+y^{2}+z^{2} ight) \ psi_{0} & =frac{sqrt{2} z}{A} Rleft(x^{2}+y^{2}+z^{2} ight) \ psi_{-1} & =frac{1}{A}(x-i y) Rleft(x^{2}+y^{2}+z^{2} ight) end{aligned} ight. ] where ( A ) is a normalization factor and ( R ) is a certain well-behaved function of ( r^{2}=x^{2}+y^{2}+z^{2} ) (you can check for the same normalization factor, if one wave function is normalized and the other two are also normalized) (a) Show that the three wavefunctions are orthogonal to each other, i.e., ( int psi_{m}^{} psi_{m^{prime}}=delta_{m, m^{prime}} ). (b) Write the three wavefunctions into spherical coordinates (c) Show that the three states are all eigenfunctions of ( hat{L}_{z}=hat{x} hat{p}_{y}-hat{y} hat{p}_{x} ), i.e., [ hat{L}_{z} psi_{m}=m hbar psi ] (d) For an operator ( hat{A} ), we define a matrix ( [hat{A}] ) with the matrix elements [ [hat{A}]_{m, m^{prime}} equiv hat{A}_{m, m^{prime}} equiv int psi_{m}^{} hat{A} psi_{m^{prime}} d x d y d z ] Using equation (8), show that the matrix form of ( hat{L}_{z} ) is

          ( -hbar ), an ( L^{2}=2 hbar ).
Consider three wavefunctions
[
left{egin{aligned}
psi_{+1} & =frac{-1}{A}(x+i y) Rleft(x^{2}+y^{2}+z^{2}
ight) \
psi_{0} & =frac{sqrt{2} z}{A} Rleft(x^{2}+y^{2}+z^{2}
ight) \
psi_{-1} & =frac{1}{A}(x-i y) Rleft(x^{2}+y^{2}+z^{2}
ight)
end{aligned}
ight.
]
where ( A ) is a normalization factor and ( R ) is a certain well-behaved function of ( r^{2}=x^{2}+y^{2}+z^{2} ) (you can check for the same normalization factor, if one wave function is normalized and the other two are also normalized)
(a) Show that the three wavefunctions are orthogonal to each other, i.e., ( int psi_{m}^{*} psi_{m^{prime}}=delta_{m, m^{prime}} ).
(b) Write the three wavefunctions into spherical coordinates
(c) Show that the three states are all eigenfunctions of ( hat{L}_{z}=hat{x} hat{p}_{y}-hat{y} hat{p}_{x} ), i.e.,
[
hat{L}_{z} psi_{m}=m hbar psi
]
(d) For an operator ( hat{A} ), we define a matrix ( [hat{A}] ) with the matrix elements
[
[hat{A}]_{m, m^{prime}} equiv hat{A}_{m, m^{prime}} equiv int psi_{m}^{*} hat{A} psi_{m^{prime}} d x d y d z
]

Using equation (8), show that the matrix form of ( hat{L}_{z} ) is

$( -hbar ), an ( L^2=2 hbar ). Consider three wavefunctions [ lefteginaligned psi+1 =frac-1A(x+i y) Rleft(x^2+y^2+z^2 ight) psi0 =fracsqrt2 zA Rleft(x^2+y^2+z^2 ight) psi-1 =frac1A(x-i y) Rleft(x^2+y^2+z^2 ight) endaligned ight. ] where ( A ) is a normalization factor and ( R ) is a certain well-behaved function of ( r^2=x^2+y^2+z^2 ) (you can check for the same normalization factor, if one wave function is normalized and the other two are also normalized) (a) Show that the three wavefunctions are orthogonal to each other, i.e., ( int psim^* psim^prime=deltam, m^prime ). (b) Write the three wavefunctions into spherical coordinates (c) Show that the three states are all eigenfunctions of ( hatLz=hatx hatpy-haty hatpx ), i.e., [ hatLz psim=m hbar psi ] (d) For an operator ( hatA ), we define a matrix ( [hatA] ) with the matrix elements [ [hatA]m, m^prime equiv hatAm, m^prime equiv int psim^* hatA psim^prime d x d y d z ] Using equation (8), show that the matrix form of ( hatLz ) is$

Added by Zhiheng L.

Question

Please give Ace some feedback