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Quantum Mechanics Simulation: Wave Functions and Hamiltonian Computation

This project involves the simulation of quantum mechanical wave functions for various energy levels (n=1 to n=5) using the Hamiltonian in a one-dimensional infinite potential well. Utilizing numerical methods, we calculate the eigenvalues and eigenfunctions, represented in an optimized format. The program sets parameters for system size and discretization, computes Hamiltonian matrix elements, and outputs the wave functions and perturbed energies for each state. This study enhances understanding of quantum states in confined systems.

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Quantum Mechanics Simulation: Wave Functions and Hamiltonian Computation

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  1. Project 1 results nmax=100

  2. Dependence on nmax

  3. Wave function n=1

  4. Wave function n=2

  5. Wave function n=3

  6. Wave function n=4

  7. Wave function n=5

  8. Specification part implicit none INTEGER, PARAMETER :: nmax=100 INTEGER, PARAMETER :: nbin=1000 REAL*8, PARAMETER :: L=10.0d0 REAL*8 :: pi,fac REAL*8 , dimension(nmax) :: w,fv1,fv2,wpert,e0 REAL*8 , dimension(nmax,nmax) :: hamil,z REAL*8 :: psi,x,dx INTEGER :: matz,ierr,m,n,nb pi=4.0d0*datan(1.0d0) hamil=0.0d0 wpert=0.0d0 dx=L/nbin

  9. Execution part do m=1,nmax e0(m)=(pi*m)**2/(2.0d0*L**2) enddo do m=1,nmax hamil(m,m)=e0(m) do n=1,nmax fac=-(3.0d0/(2.0d0*L))*dsqrt(2.0d0*pi/L)* * (dcos(pi*(m-n)/2.0d0)*dexp(-(n-m)**2*pi**2/(2.0d0*L**3))- * dcos(pi*(m+n)/2.0d0)*dexp(-(n+m)**2*pi**2/(2.0d0*L**3))) hamil(m,n)=hamil(m,n)+fac if(m.eq.n) wpert(m)=wpert(m)+hamil(m,m) if(m.ne.n) wpert(m)=wpert(m)+fac**2/(e0(m)-e0(n)) enddo write(6,*) m,hamil(m,m) enddo matz=1 call rs(nmax,nmax,hamil,w,matz,z,fv1,fv2,ierr)

  10. do n=1,nmax write(6,*) w(n),wpert(n),e0(n) enddo fac=dsqrt(2.0d0/L) do m=1,5 do nb=1,nbin psi=0.0d0 do n=1,nmax x=dx*nb psi=psi+fac*z(n,m)*dsin(pi*x*n/L) enddo write(6,*) x,psi enddo write(6,*) enddo stop end 55,2-9 4%

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