Question

Consider a particle in a ( 1 mathrm{D} ) infinite square well with walls at ( x=pm L / 2 ). The eigenstates for this potential are [ psi_{n}(x)=langle x mid n angle=left{egin{array}{ll} sqrt{frac{2}{L}} cos (pi n x / L), & n= ext { odd } \ sqrt{frac{2}{L}} sin (pi n x / L), & n= ext { even } end{array} ight. ] with energies ( E_{n}=frac{pi^{2} hbar^{2}}{2 m L^{2}} n^{2} ) ( ( n ) is an integer). A small additional potential ( L V_{0} delta(x) ) is now applied. a. Use perturbation theory to show that to first order in ( V_{0} ) the energies of the ( n= ) odd states are increased by ( 2 V_{0} ) while the energies of the ( n= ) even states are unchanged. b. What is the shift in energy of the ( n= ) even states to second order in ( V_{0} ) ?

          Consider a particle in a ( 1 mathrm{D} ) infinite square well with walls at ( x=pm L / 2 ). The eigenstates for this potential are
[
psi_{n}(x)=langle x mid n
angle=left{egin{array}{ll}
sqrt{frac{2}{L}} cos (pi n x / L), & n=	ext { odd } \
sqrt{frac{2}{L}} sin (pi n x / L), & n=	ext { even }
end{array}
ight.
]
with energies ( E_{n}=frac{pi^{2} hbar^{2}}{2 m L^{2}} n^{2} ) ( ( n ) is an integer).
A small additional potential ( L V_{0} delta(x) ) is now applied.
a. Use perturbation theory to show that to first order in ( V_{0} ) the energies of the ( n= ) odd states are increased by ( 2 V_{0} ) while the energies of the ( n= ) even states are unchanged.
b. What is the shift in energy of the ( n= ) even states to second order in ( V_{0} ) ?

$Consider a particle in a ( 1 mathrmD ) infinite square well with walls at ( x=pm L / 2 ). The eigenstates for this potential are [ psin(x)=langle x mid n angle=lefteginarrayll sqrtfrac2L cos (pi n x / L), n= ext odd sqrtfrac2L sin (pi n x / L), n= ext even endarray ight. ] with energies ( En=fracpi^2 hbar^22 m L^2 n^2 ) ( ( n ) is an integer). A small additional potential ( L V0 delta(x) ) is now applied. a. Use perturbation theory to show that to first order in ( V0 ) the energies of the ( n= ) odd states are increased by ( 2 V0 ) while the energies of the ( n= ) even states are unchanged. b. What is the shift in energy of the ( n= ) even states to second order in ( V0 ) ?$

Added by Foster J.

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