Infinite square well potential
The particle of mass m is in one dimensional infinite square well potential characterized by
V(x) = 0 ; -L< x< L
= ∞ ; l xl ≥ L
The time independent Schrodinger equation for -L< x< L, V(x) = 0 can be written as
d²u(x)/ dx² + 2mE u(x)/ ℏ² = 0....i
where u(x) represents the wave functio.
We have
k² = 2mE/ℏ² ......ii
Eqn i becomes
d²u(x)/ dx² + k²u(x) = 0 ...iii
The general solution of this will be
u(x) = A cos kx + B sin kx ...iv
The condition l x l ≥ L ; v(x)-> ∞ satisfies the condition of bound state. For bound state, the boundary condition is
l u(x) l² -> 0 for lxl -> L
That means wave functions vanishes at x = ± L
Thus for x = -L, u(x) = 0
Applying this in the equation iv,
0 = A cos k(-L) ∓ B sin k(-L)
A cos kL - B sin kL = 0 ....v
Similarly for x = + L , u(x) = 0
Eqn iv becomes
0 = A cos kL + B sin kL .....vi
Adding v and vi we get
2 A cos kL = 0
cos kL = 0
cos kL = cos n𝜋/ 2 for n = 1, 3, 5....
Similarly, subtracting vi and v
2B sin kL = 0
sin kL = sin n 𝜋/ 2 for n= 2,4,6....
k = n 𝜋/ 2L for even n ......vii
Thus for both odd and even n
k = n 𝜋/ 2L
Substituting this in equation i
( n 𝜋/ 2L)² = 2mEn/ ℏ²
En = n²π²ℏ²/ 8mL² .........ix
Equation ix gives the energy spectrum of the particle inside square well potential. It is found that the energy spectrum is discrete as n =1,2,3,... are as follows
E_1 = E
E_2 = 4E
E_3 = 9E
For odd n, the wavefunction can be expressed using iv
U_odd (x) = A cos (nπ/ 2L) x .....x
Similarly for even n,
U_even (x) = B sin (nπ/ 2L) x .....xi
Equation x and xi can be normalized using the normalized condition
Let us normalized equation xi
Similarly normalizing equation x yield
A = 1 / \sqrt{L}
Thus, the normalized eigen functions of the particle are
u_even (x) = 1 / \sqrt{L} sin (nπ/ 2L) x ..... xii
u_odd (x) = 1 / \sqrt{L} cos (nπ/ 2L) x ..... xiii
The parity of eigen functions are as follows
u_even (x) = 1 / \sqrt{L} sin (nπ/ 2L) x = - u_even (x) ....xiv
Even eigen function has an odd parity
Similarly,
u_even (x) = 1 / \sqrt{L} cos (nπ/ 2L) x = u_odd (x) ....xv
Odd eigen function has even parity
The energy interval between two successive levels
E_n+1 - E_n = π²ℏ²/ 8mL² [ (n+1)² - n² ]
when L-> ∞, E_n+1 - E_n becomes zero, this means the discrete spectrum becomes continuous spectrum.
This note is a part of the Physics Repository.