Quantum Mechanics (Practical)

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### Topics

• Finding Eigenstates Solving Transcendental Equation: To find eigenvalues of the bound state particle of mass in a one dimensional potential well by solving the transcendental equation that appears as the eigenvalue condition (graphs are to be plotted for appropriate guess values, scipy root searching package may be used) and to plot the eigenfunctions.
• Use of Shooting Algorithm: Shooting algorithm for solving bound state problems (solving the ode using both Euler and Numerov algorithms) : conversion to dimensionless variable, eigenvalues and eigenvectors of the ground and first excited states.
• in one dimension (for example, the Harmonic oscillator, the Morse potential, the triangular well etc.)
• the s-wave radial equation for a particle moving in a central potential, d2U(r)⁄dr2 = A(r)U(r) where A(r) = 2m ⁄ ℏ2 [V(r)-E]. Some examples.
• V(r) = - e2⁄r.
• V(r) = - e2⁄r e-r/a .
• V(r) = 1⁄2 kr2 + 1⁄3 br3.
• V(r) = D(e-2αr' - e-αr'), where r'=r-r0⁄r.
• Time Evaluation of Wave Packet:
• Time evolution of a wave packet moving in free space by the numerical solution of the time dependent ScrÃ¶dinger equation.
• Solving the TDSE to study Barrier penetration and tunneling for an initially Gaussian wavepacket.

### Remarks

• Will be updated from time to time.

### Study Materials

• An Introduction to computational Physics - T.Pang.
• Scientific Computing in Python - Abhijit Kar Gupta.
• Computational Physics problem solving with Computers - Landau, Paez, Bordeianu.
• Computational Methods for physcs - Joel Franklin.
• Computational Quantum Mechanics - Joshua Izaac, Jingbo Wang.