Quantum Systems

1
Quantum Systems

• In the molecular dynamics approach the classical
  trajectory of each particle is calculated as a
  function of time
• we cannot use the same approach in quantum
  systems
• quantum mechanics does not allow us to specify
  both the position and momentum of a particle
  simultaneously

2
Quantum Mechanics

• Quantum mechanics does allow us to analyze
  probabilities but there are problems

• Recall the 1-d diffusion equation
• a numerical solution requires us to make x and t
  discrete variables
• we would need about 103 points for each value of
  t whereas molecular dynamics only required one
  point

3
Quantum methods

• For many degrees of freedom the problem is more
  apparent
• P(x1,x2,,xN,t) gt need N1000 points
• methods generally limited to a small number of
  particles
• since the diffusion equation can be formulated as
  a random walk, it will be no surprise that
  Schrodingers equation can be analyzed in a
  similar way.

4
Review of Quantum Theory

• Consider a 1-d non-relativistic quantum system
• the state of the system is characterized by a
  wave function ?(x,t) which is the probability
  amplitude
• probability of finding a particle in the volume
  dx centered about x is P(x,t)dx ?(x,t) 2dx
• hence ?(x,t) must be normalized such that

5
Quantum Theory

• In the presence of an external potential V(x,t),
  the time evolution of ?(x,t) is determined by
  the time-dependent Schrodinger equation

• Each physical variable A has a corresponding
  operator Aop
• the average value of A is

6
Quantum Review

• For example xop x
• pop -i??/?x
• if the potential is independent of t, ?(x,t)
  ?(x)e-iEt/?
• substituting in the Schrodinger equation we
  obtain the time-independent equation

7
Quantum Review

• In general there are many eigenfunctions ?n and
  corresponding eigenvalues En
• hence ?(x,t) ?cn ?n e-iEnt/ ?
• for bound state solutions E lt V the allowed
  energies are quantized
• ?n(x) must be finite for all x and bounded for
  large x so that it can be normalized
• numerical solution?

8
Numerical Solution

• Divide the range of x into intervals of width ?x
• denote discrete positions as xs s ?x
• denote ?s ?(xs) and ?s ?(xs)
• if the potential V(x) satisfies V(-x)V(x) then
  the solution have definite parity even or odd
• choose ?(0)1 and ?(0)0 for even parity
• choose ?(0)0 and ?(0)1 for odd parity
• guess a value for E
• compute ?s1 and ?s1 using the algorithm

9
Algorithm

• iterate ?(x) toward increasing x until ?(x)
  diverges
• change E and repeat the steps
• bracket the value of E which gives an acceptable
  solution by watching for divergences in opposite
  directions

10
Square Well Potential

• V(x) 0 for x lt a V0 for x gt
  a
• need to input V0, a, parity, initial guess for E
  , step size ?x and xmax the maximum value of x
• try V0150, a1, ?x .01, xmax 4

eigen
11
Time-dependentSchrodinger Equation

• Numerical solution of the time-independent
  equation is straightforward
• constant energy solutions do not require us to
  make time discrete
• how would we solve the time-dependent equation?
• Naïve approach would be to produce a grid in the
  x-t plane
• tnt0n ?t xsx0s ?x ?(x,t) gt ?(xs,tn)

12
Algorithms

• Need an algorithm that relates ?(xs,tn1) to the
  value of ?(xs,tn) for each value of xs
• an example is

• This algorithm leads to unstable (divergent)
  solutions

13
Algorithms

• A better approach treats the real and imaginary
  parts of ? separately
• this algorithm ensures that the total probability
  remains constant

• The Schrodinger equation becomes (?1)

14
Algorithm

• Numerical solution of these equations is based on

• The probability density is conserved if we use

15
Initial Wavefunction

• Consider a Gaussian wave packet

• The expectation value of the initial velocity is
  ltvgtp0/m ?k0/m
• in the simulation set m ?1

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