Question

Determine for which values of m the function variant phi (x )equals x Superscript m is a solution to the given equation.

Answer 1

Quantum numbers n, l, and m define an electron's state in an atom. The magnetic quantum number m spans -l to +l for a given l, which is determined by n-1. For example, if n = 2 and l = 1, then m can be +1, 0, or -1.

Step-by-step explanation:

Determination of Allowed Quantum Numbers

The question seems to be rooted in the quantum mechanical description of atomic structures, specifically with respect to Schrödinger's equation and the associated quantum numbers. When interpreting wave functions in quantum mechanics, each state is defined by quantum numbers n, l, and m. For a given value of the azimuthal quantum number l, the magnetic quantum number m can take on values ranging from -l to +l, including zero. This determines the orientation of the orbital's angular momentum. For l = 1, m could be +1, 0, or -1. Each of these corresponds to different magnetic sublevels within an atomic orbital.

As for the quantum number n, which represents the principal quantum number, it defines the energy level of an electron in an atom and indirectly the size of the orbital. The magnetic quantum number m₁ has allowed values that are dependent on the azimuthal quantum number l, not directly on n. However, the maximum value of l is determined by n-1. Therefore, the values of m₁ are constrained by the chosen value of n through the value of l.

For example, if n = 2, we know that l can be 0 or 1, and accordingly, m₁ can be 0 for l = 0, and +1, 0, -1 for l = 1.

Answer 2

1)
`m=-3\,;\,\, m=(1)/(3)`

2)
`m=1\pm√(6)\\\\`

Step-by-step explanation:

As differential equation is not give so we consider both equations attached in figure below one by one.

`1) \,3x^(2)(d^2y)/(dx^2)+11x(dy)/(dx)-3y=0`

Writing in terms of ∅

`3x^(2)(d^2\phi)/(dx^2)+11x(d\phi)/(dx)-3\phi=0`

Substitute

`\phi=x^m\\\\\phi'=mx^(m-1)\\\\\phi''=m(m-1)x^(m-2)`

`3x^(2)(m(m-1)x^(m-2))+11x(mx^(m-1))-3x^m=0\\\\3x^(m)(m(m-1))+11mx^(m)-3x^m=0\\\\x^(m)(3m^2-3m+11m-3)=0\\\\3m^2+8m-3=0\\\\By \,\,factorization\\\\3m^2-m+9m-3=0\\\\m(3m-1)+3(3m-1)=0\\\\(m+3)(3m-1)=0\\\\\implies m=-3\,;\,\, m=(1)/(3)`

`2)\, x^2(d^2y)/(dx^2)-x(dy)/(dx)-5y=0---(2)`

Writing in terms of ∅

`x^2(d^2\phi)/(dx^2)-x(d\phi)/(dx)-5\phi=0`

Substitute

`\phi=x^m\\\\\phi'=mx^(m-1)\\\\\phi''=m(m-1)x^(m-2)`

`x^2(m(m-1)x^(m-2))-x(mx\phix^(m-1))-5x^(m)=0\\\\(m^2-m)x^m-mx^m-5x^m=0\\\\x^m(m^2-m-m-5)=0\\\\m^2-2m-5=0\\\\Using\,\,quadratic\,\,formula\\\\m=(2\pm √(20+4))/(2)\\\\m=(2\pm √(24))/(2)\\\\m=1\pm√(6)\\\\`