1. Consider a particle of mass m in a one- dimensional box -L< x < +L as shown below.
The potential of the system is given as
0 -L < x < +L
V =
8 x <-L and x > +L
(a) Determine the energy eigenvalue of the particle.
(b) Show that the expectation value of the momentum

= 0.
(c) Determine the expectation value of the kinetic energy .
(d) What is the energy eigen values if the boundary of thebox
is shifted to -L < x < +L/2

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A particle of mass m is bound in a finite one-dimensional squarewell described by the potential energy function: V(x)=0 for x<a(region I)
=-v0 for -a<=x<=a (region II) with v0>0
=0 for x>a (region III)
The energy of the particle has some definite negative value-v0<E1<0.
Consider the time independent Schrodinger equation.
Question: In each of the three regions write the normalizablespatial wave function f(x) of the particle assuming that f is aneven function of x, i.e. f(x)=f(-x) and calculate the constants inyour function that are determined by the given quantities.

1. The problem statement, all variables and given/known data

A particle is in a 1D box of length L on x-axis with0<x<L.What is the shift in the energy of the energy ofeach wavefunctiondue to the weak spring potential to the firstorder k? The answerinvolves an integral found on an integraltable.

2. Relevant equations

H=p²/2m
Si_n=sqrt(2/L)cos(nÏx/L)
E_n=n²Ï²hbar²/2mL²
V(x)=kx²/2

3. The attempt at a solution

I'm super confused because the shift in the energy isusuallydetermined by a sum, not an integral. Unless IcalculateE_n⁽¹⁾ by taking;

E_n⁽¹⁾=<n|H|n>
=<n|V|n>
=V * integral(2/Lsin²(nxpi/L)dx) with limits 0 toL

Is this the correct way to do this one? I didn't include kvery wellunless I put it in for V at the end...

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