A particle is confined in a 1-dimensional box of at the region [0,L]. At time 𝑡=0 it is known to be in the state( , 0) sin( / )x t N n x L .(a) Find N by normalizing the wavefunction.
Question
A particle is confined in a 1-dimensional box of at the region [0,L]. At time 𝑡=0 it is known to be in the state( , 0) sin( / )x t N n x L .(a) Find N by normalizing the wavefunction.
Solution
The normalization condition for the wave function is given by the integral over all space of the absolute square of the wave function, which must equal 1. In mathematical terms, this is:
∫|ψ(x, 0)|² dx = 1, where the integral is from 0 to L.
Substituting the given wave function into this equation gives:
∫(from 0 to L) |N sin(nπx/L)|² dx = 1.
Squaring the absolute value of the wave function gives:
∫(from 0 to L) N² sin²(nπx/L) dx = 1.
The integral of sin²(nπx/L) from 0 to L is L/2. Therefore, the equation becomes:
N² * L/2 = 1.
Solving for N gives:
N = sqrt(2/L).
So, the normalization constant N is sqrt(2/L).
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