The legendrian operator is given as \\(\\Lambda^{2} = \\frac{1}{sin\\theta}\\fra

Question

The legendrian operator is given as

   (Lambda^{2} = rac{1}{sin heta}rac{partial}{partial heta}(sin hetarac{partial}{partial heta}) + rac{1}{sin^{2} heta}rac{partial^{2}}{partial phi^{2}})   

One eigenfunction of the legendrian operator is given as    (Y_{2,+2}( heta,phi) = sqrt{rac{15}{32pi}}*sin^{2} heta*e^{+2iphi})   

Write the eigenvalue eqation (which should just be the Schrodinger Equation without the h bar squared term) and calculate the corresponding eigenvalue. You will need to use differentiation.

Explanation / Answer

Eigenvalue Equation:

lambda^2 (Y 2,+2) = E * (Y 2,+2)

Here, lambda ^2 = legendrian operator, E = eigenvalue

Now, lambda ^ 2 (Y 2,+2) =

lambda^2 ( sqrt(15/32pi) * sin^2(theta) * exp(2i*phi))

= 1/sin(theta) * del/del(theta) ( sqrt(15/32pi) *exp(2i*phi) * 2 (sin(theta))^2*cos(theta)) + sqrt(15/32pi) * (-4)exp(2i*phi)

= sqrt(15/32pi) * exp(2i*phi) * [ 4(cos(theta))^2 - 2(sin(theta))^2 -4]

= sqrt(15/32pi) * exp(2i*phi) * [ -6 (sin(theta))^2]

= -6 * (Y 2,+2)

So Eigenvalue E = -6

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