Particle in 1D Potential Well (Solving Transcendental Equation): Program 1

'''To find eigenvalues of the bound state particle of mass in a one
dimensional potential well by solving thetranscendental equation
that appears as the eigenvalue condition and to plot the eigenfunctions.'''

import numpy as np
from scipy.integrate import odeint, simps
import matplotlib.pyplot as plt
from scipy import optimize as opt

def evenparity(z, z0):
    return np.tan(z) - np.sqrt((z0/z)**2 - 1)

def oddparity(z, z0):
    return 1.0/np.tan(z) + np.sqrt((z0/z)**2 - 1)

# Potential parameters; W=width, D=depth 
L = 4; V0 = 10; h_bar = 1; m = 1;
Erange = np.arange(-V0+0.001, 0, 0.001)
x = np.linspace(-L, L, 1000)

# ND parameter
z = (L/2)*np.sqrt(2*m*(V0+Erange))/h_bar
z0 = (L/2)*np.sqrt(2*m*V0)/h_bar

plt.subplot(121)
plt.plot(z, np.tan(z), '--k+', lw=.4, label="tan(z)")
plt.plot(z, np.sqrt((z0/z)**2-1), '--r.', lw=.4, label = r"$\sqrt{(\frac{z_0}{z})^2-1}$")
plt.title("Finite Square Well (Even Parity)")
plt.legend()
plt.xlabel("z", size=16)
plt.ylabel(r"$f(z)$", size=16)
plt.ylim([-15, 15])

plt.subplot(122)
plt.plot(z, 1.0/np.tan(z), '--bx', lw=.4, label="cot(z)")
plt.plot(z, -np.sqrt((z0/z)**2-1), '--g>', lw=.4, label = r"-$\sqrt{(\frac{z_0}{z})^2-1}$")
plt.title("Finite Square Well (Odd Parity)")
plt.legend()
plt.xlabel("z", size=16)
plt.ylabel(r"$f(z)$", size=16)
plt.ylim([-15, 15])
plt.tight_layout()
plt.savefig('plot.png')
plt.cla()

# Appropriately guess the roots from graph
evenroot_guess = [1.36, 4.2, 7.01]
oddroot_guess  = [2.8, 5.57, 8.19]

# Compute z and check the authenticity of the roots
evenroot = opt.root(evenparity, evenroot_guess, args = (z0, ))
oddroot =  opt.root(oddparity,  oddroot_guess,  args = (z0, ))
evE = (evenroot.x*h_bar*2/L)**2/(2.0*m)-V0
oddE = (oddroot.x *h_bar*2/L)**2/(2.0*m)-V0
print('Energy eigen values of even states:',evE)
print('Energy eigen values of odd states:', oddE)
E1 = np.append(evE, oddE)
E = np.sort(E1)

def V(x):
    if(x > -L/2 and x < L/2):
        V = -V0
    else:
        V = 0
    return V

P = np.array([V(i) for i in x])

def f(u, x, E):
    y, z = u
    f1, f2 = z, ((2*m)/(h_bar)**2)*(V(x) -E)*y
    return (f1, f2)

u = [0, 0.015]
plt.figure(figsize=(10,20))
st = 1
for eigen in E:
    sol = odeint(f, u, x, args = (eigen, ))
    psi = sol[:, 0]
    normpsi = eigen + psi/np.sqrt(simps(psi*psi, x))
    plt.subplot(len(E) , 2, st)
    plt.ylabel(r"$\Psi$"+str(st)+"(x)", fontsize=16)
    plt.xlabel(r'$x$', fontsize=16)
    plt.title('Energy eigen value = '+str(eigen))
    plt.plot(x, normpsi)
    plt.plot(x, P)
    plt.axhline(y = eigen, color = 'g', lw = 0.5)
    plt.axvline(x = 0, color = 'g', lw = 0.5)
    st = st + 1
plt.tight_layout()
plt.savefig('psi.png')
plt.show()

Output 1

The provided Python code is designed to solve the eigenvalue problem for a particle bound in a one-dimensional potential well. The code calculates the energy eigenvalues and plots the corresponding eigenfunctions. Below is an explanation of the code, broken down into its components:

1. Imports

numpy: Provides support for array operations.
scipy.integrate.odeint: Used to solve ordinary differential equations (ODEs).
scipy.integrate.simps: For numerical integration using Simpson's rule.
matplotlib.pyplot: For plotting graphs.
scipy.optimize: For finding roots of transcendental equations.

2. Functions for Parity Conditions

evenparity(z, z0): Represents the transcendental equation for even parity states.
oddparity(z, z0): Represents the transcendental equation for odd parity states.
These functions are used to find the eigenvalues, corresponding to the points where these functions equal zero.

3. Potential Parameters and Energy Range

L = 4: Width of the potential well.
V0 = 10: Depth of the potential well.
h_bar = 1, m = 1: Reduced Planck's constant and mass of the particle, respectively.
Erange: Array representing the range of energies considered, from just above -V0 to 0.
x: Array representing spatial coordinates.

4. Dimensionless Parameters

z: Dimensionless quantity related to the energy and well width.
z0: Dimensionless quantity related to the well depth.

5. Plotting the Transcendental Equations

The code plots the functions for tan(z) and cot(z) against sqrt((z0/z)^2 - 1) to visualize the transcendental equations.
These plots help in visually identifying the approximate locations of the eigenvalues.

6. Guessing Roots from the Graphs

evenroot_guess, oddroot_guess: Initial guesses for the roots (eigenvalues) based on the plots.
opt.root: Function to compute the roots of the parity conditions (i.e., the eigenvalues).

7. Energy Eigenvalues

The roots found are converted back to physical energy values using the relation E = (z*h_bar*2/L)**2/(2*m) - V0.
The computed eigenvalues for even and odd parity states are printed.

8. Potential Function V(x)

A step function defining the potential well:

Inside the well (-L/2 < x < L/2), the potential is -V0.
Outside the well, the potential is 0.

9. Solving the Schrödinger Equation

f(u, x, E): Defines the system of ODEs for the wavefunction psi(x) and its derivative psi'(x).
odeint: Solves the ODE for each eigenvalue E to obtain the wavefunction psi(x).

10. Normalization and Plotting

The wavefunctions are normalized using Simpson's rule for integration.
The code then plots the normalized wavefunctions alongside the potential well for each eigenvalue.
Each subplot corresponds to an eigenfunction, showing the wavefunction shape and energy level.

11. Output

The plots of the eigenfunctions are saved as psi.png, and the plots of the transcendental equations are saved as plot.png.

Purpose and Summary

Goal: To find the eigenvalues of a particle bound in a one-dimensional potential well and to plot the corresponding eigenfunctions.
Method: Solves the transcendental equation that governs the eigenvalue condition, then solves the Schrödinger equation to obtain the wavefunctions.
Output: Plots of the transcendental equation, energy eigenvalues, and normalized eigenfunctions.

Particle in 1D Potential Well (Solving Transcendental Equation): Program 2

import numpy as np
from scipy.integrate import odeint, simps
import matplotlib.pyplot as plt
from scipy.optimize import root

# Functions for parity conditions
def even_parity(z, z0):
    return np.tan(z) - np.sqrt((z0/z)**2 - 1)

def odd_parity(z, z0):
    return 1.0/np.tan(z) + np.sqrt((z0/z)**2 - 1)

# Potential well parameters
L, V0 = 4, 10
h_bar, m = 1, 1
x = np.linspace(-L, L, 1000)

# Dimensionless parameters
z = (L/2) * np.sqrt(2 * m * (V0 + np.arange(-V0+0.001, 0, 0.001))) / h_bar
z0 = (L/2) * np.sqrt(2 * m * V0) / h_bar

# Approximated roots for even and odd parity states
even_root_guess = [1.36, 4.2, 7.01]
odd_root_guess  = [2.8, 5.57, 8.19]

# Compute the roots
even_root = root(even_parity, even_root_guess, args=(z0,))
odd_root = root(odd_parity, odd_root_guess, args=(z0,))

# Calculate energy eigenvalues
evE = (even_root.x * h_bar * 2 / L)**2 / (2.0 * m) - V0
oddE = (odd_root.x * h_bar * 2 / L)**2 / (2.0 * m) - V0
E = np.sort(np.append(evE, oddE))

# Potential function
def V(x):
    return -V0 if -L/2 < x < L/2 else 0

P = np.array([V(i) for i in x])

# Function to solve Schrödinger equation
def schrodinger_eq(u, x, E):
    y, z = u
    return z, (2 * m / h_bar**2) * (V(x) - E) * y

# Initial condition for wavefunction
u = [0, 0.015]

# Plot eigenfunctions
plt.figure(figsize=(8, 10))
for i, eigen in enumerate(E, 1):
    sol = odeint(schrodinger_eq, u, x, args=(eigen,))
    psi = sol[:, 0]
    norm_psi = psi / np.sqrt(simps(psi**2, x))
    
    plt.subplot(len(E), 1, i)
    plt.plot(x, norm_psi, label=f'E = {eigen:.2f}')
    plt.axhline(y=eigen, color='r', linestyle='--', lw=0.5)
    plt.axvline(x=0, color='r', linestyle='--', lw=0.5)
    plt.title(f'Energy Eigenvalue = {eigen:.2f}')
    plt.legend()

plt.tight_layout()
plt.savefig('psi.png')

TISE Solution

Particle in 1D Potential Well (Solving Transcendental Equation): Program 1

Output 1

1. Imports

2. Functions for Parity Conditions

3. Potential Parameters and Energy Range

4. Dimensionless Parameters

5. Plotting the Transcendental Equations

6. Guessing Roots from the Graphs

7. Energy Eigenvalues

8. Potential Function V(x)

9. Solving the Schrödinger Equation

10. Normalization and Plotting

11. Output

Purpose and Summary

Particle in 1D Potential Well (Solving Transcendental Equation): Program 2

Output 2