References
  • MHJ Chapter 11
  • Giordano & Nakanishi Chapter 10
  • R. Landau et al Chapter 15
           Many Body System
   Schrödinger’s equation
   for a system of A nucleons (A = N + Z), N being the number of neutrons and Z the 
   number of protons). There are:
    coupled second-order differential equations in 3A dimensions. For a nucleus like 
    10Be this number is 215040. This is a truly challenging many-body problem.
    Eq. (11.1) is a multidimensional integral. As such, Monte Carlo methods are ideal 
    for obtaining expectation values of quantum mechanical operators.
    Our problem is that we do not know the exact Wave function 
    Psi(r1, .., rA, α1, .., αN).
    Our goal in this chapter is to solve the problem using  
    Variational Monte Carlo approach to quantum mechanics.
    We limit the attention to the simple Metropolis algorithm, without 
    the inclusion of importance sampling. Importance sampling and 
    diffusion Monte Carlo methods are discussed in chapters
    18 and 16.
    Before that we give a short review about Quantum Mechanics in the 
    next slides
    Postulates of Quantum Mechanics
   Schrödinger’s equation for a one-dimensional one body problem: 
    if we perform a rotation of time into the complex plane, using τ = it/hbar, the 
    time-dependent Schrödinger equation becomes
    With V = 0 we have a diffusion equation in complex time!
   The diffusion constant:
    The wave function have to satisfy: