Question

# The quantum state of a particle can be specified by giving a complete set of quantum...

The quantum state of a particle can be specified by giving a complete set of quantum numbers (n,l, ml,ms). How many different quantum states are possible if the principal quantum number is n = 2?

To find the total number of allowed states, first write down the allowed orbital quantum numbers l, and then write down the number of allowed values of ml for each orbital quantum number. Sum these quantities, and then multiply by 2 to account for the two possible orientations of spin.

Given

n = 2 as principle quantum number

we know that the orbital quantum number is equal to n that from 0 to n-1 so

the allowed orbital quantum numbers l are , 0,(2-1) ==> 0,1

the number of allowed values of ml values 2l+1 , m = 2*0+1 ,2*1+1 = 1,3

the spin values are s = +1/2 ,-1/2

the total number of is N = sum of (2(2l+1)) = 2((2*0+1)+(2*1+1)) = (2((1)+(3)) = 2*4 = 8

that is the possible quantum states for principle quantum number n is

N = 2*n^2 = 2*2^2 = 2*4 = 8

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