Computational Methods in Quantum Mechanics

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Computational Methods in Quantum Mechanics

Branislav K. Nikolić

Department of Physics & Astronomy, University of Delaware, Newark, DE 19716, U.S.A.

PHYS 460/660: Computational Methods of Physics

http://wiki.physics.udel.edu/phys660

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Quantum Mechanics

in the Words of the Founding Fathers

Albert Einstein: “Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the

secret of the Old One. I am at all events convinced that He does not play dice .”

 Niels Bohr: “Anyone who is not shocked by quantum theory has not understood a single word.”

 Werner Heisenberg: “I myself . . . only came to believe in the

uncertainty relations after many pangs of conscience. . . “

 Erwin Schrödinger: “Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business!”

 Groucho Marx: ”Very interesting theory - it makes no sense at all.”

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Quantum Interference in Two-Slit Experiments

 The self interference of individual particles is the greatest mistery in quntum physics; in fact, Richard Feynman pronounced it “the only mystery” in quantum theory

 

1 2

2 2 2 *

1 2 1 2

1 2

1 2 Re

prob 2

    

 

         

 

1 2

2 2

1 2

1 1 2

2 1

prob 2  

      

 

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Quantum Erasure

B. Green, Fabric of Cosmos (page 149):“These experiments are a magnificent affront to our conventional notions of space and time. Something that takes place long after and far away from something else nevertheless is vital to our description of that something else. By any classical-common sense-reckoning, that's, well, crazy. Of course, that's the point: classical reckoning isthe wrong kind of reckoning to use in a quantum universe.... For a few days after I learned of these experiments, I remember feeling elated. I felt I'd been given a glimpse into a veiled side of reality.

Common experience—mundane, ordinary, day-to-day activities—

suddenly seemed part of a classical charade, hiding the true nature of our quantum world. The world of the everyday suddenly seemed nothing but an inverted magic act, lulling its audience into believing in the usual, familiar conceptions of space and time, while the astonishing truth of quantum reality lay carefully guarded by nature's sleights of hand.”

 

0

1 path1 path 2

  2     

 

0

1 path1 path 2 2

1 path1 path 2

     

  

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Classical vs. Three Roads to Quantum Mechanics

Hamilton-Lagrange Mechanics

i

,

i

i i

p q

q p



  



 

 

H H

 ^,  ⁰ ^{( , )}

^{( , )}

t

H p q

d e

dt p q Z

  ^       

^

 H

Dirac-Heisenberg-Schrödinger Textbook Formulation in Hilbert Space

Moyal-Groenewald-Weyl Deformation Quantization

Feynman-Dirac Path Integral

, A ˆ

ˆ

ˆ ˆ

A A [ , ] ˆ ˆ

i H

t

i d A H

dt t



   



  





[ ]/

(

_f, _f _i

, )

_i

( )

^{iS q}

K q t  q t   Dq t e

^

( )/2

( , )

1( * )

q p p q

W

W W

i

q p

H H

t i e



  

   

   



 ^ ^{ } ^{ }



P Phase Orbit

P Feynman Paths Boltz

mann-Gibbs Statistical M

echanics

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Representation of Quantum Mechanics: Usual Textbook Ones Cannot Be Put on the Computer

 Coordinate Representation:

 Momentum Representation:

 Energy representation:

( )

x x dx x x dx

      

/ /

1 1

( ) ( )

2 2

ipx ipx

p p dp p p x x dpdx

p x e p e x dx

 

 

    

    

 



 

n n n n

E E E

       ˆ n n n

H E  E E

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Quantum-Mechanical Probabilities in Practice

 Probability to find coordinate of a particle in

 Probability to find momentum of a particle in

 Probability to find energy of a particle to be

* *

* 2

( ) ( ) ( ) ( ) 1

ˆ , ( ) ( ) ( )

x x dx x x dx x x dx

x x x x x x x





        

    



2 * *

| | 1

n n n n n n n

n

E E E

           

[ , x x  dx ]

[ , p p  dp ]

* *

* 2

( ) ( ) ( ) ( ) 1

ˆ , ( ) ( ) ( )

p p dp p p dp p p dp

p p p p p p p





        

    



E n

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Schrödinger Equation(s)

 time evolution of state vectors:

 time evolution of coordinate wave functions for a single particle acted on by a conservative force with potential

 Momentum representation for a single particle of mass m acted on by a conservative force with potential

ˆ stationary: ˆ

i d H H E

dt

      



2 2

( ) ( ) ( ) ( )

i 2 V

t m

      



x  x x x



( ) V x

( ) 2

( ) ( ') ( ') '

i 2 p V d

t m





     

 ^p ^p  ^{p p} ^p ^p



1

/

( ) ( ) ( )

2 V x V p V x e

ipx

dx



  _ 

 ^

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Heisenberg Uncertainty Relations

 Operators (of course, Hermitian) which represent physical quantities in QM in general do not commute:

 Coordinate and momentum are c-numbers in classical physics, but in QM:

 

²

 

² ² ²

ˆ , ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ 1 ˆ , ˆ 2

ˆ ˆ ˆ ˆ ˆ ˆ 0 iff ˆ

A B A B B A A B A B

A A A A A A A a

            

   

          

 ^{ˆ ˆ} ^,  ^{ˆ ˆ}

x x

2 x p  i    x p  



0 xp  px 

2 x  2 p 

 x  _p

The experimental test of the Heisenberg inequality does not involve simultaneous measurements of x and p, but rather it involves the measurement of one or the other of these dynamical variables on each independently prepared representative of the particular state being studied.



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Quantum Tunneling Through

Single Barrier in Solid State Systems

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Quantum Tunneling Through Double Barrier in Textbooks

, ,

( ) ,

, ,

m m

ikx ikx

q x q x

ik x ik x

p x p x

ikx ikx

A e B e x a

C e D e a x b

x F e G e b x c

H e Ie c x d

Je K e x d



  

   

 

     

   

   



1

2 2

2

2 2

2 ( )

2 ,

2 ( ) 2 ( )

,

m m

m V E

k m E q

m E V m V E

k p

  

 

 

 

*

( ) ( )

x

2 x x

j x x

mi x x

   

         



2

for electrons incident from the left (+ states) 2

, , 2

for electrons incident from the right (- states) 2

| |

| | ,

| |

k transmitted x d

incident x a

k

H

j A

T j B

K







  



2 2

2

( ) ( ) ( ) 2

d x

E V x

m dx

     

Transmission probability versus incident energy for:

(a)asymmetric barriers; (b) symmetric barriers; (c) tall barriers; and (d)narrow well.

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Quantum Tunneling Through

Double Barrier in Solid State Systems

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Plane Wave Solutions of the Free Particle Schrödinger Equation

 ¹ 

_{1 2}

 

²

( ) exp ,

2 2

p

i p

x Et px E

 m

 

          

 Plane waves are solution of the free particle Schrödinger equation in coordinate representation:

2 2

( )

2

( , )

p

2

p

i t x t

t m x

 

   

 

 

   

2 2 2

1 2 1 2

2 1 1

e e

2 2 2 2

i i

px p px

m x  m 

  

 ^ ^



 

H ˆ E E E

 Plane waves are eigenstates of the free particle Hamiltonian:

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Construction of Gaussian Wave Packet From Plane Waves

 ¹  _{1 2}   ²

( ) exp ,

2 2

p

i p

x Et px E

 m

 

          

 Linear superpositions of plane waves are also solutions!

( , ) _n _p ( , ) ( , ) ( ) _p ( 0 , )

n

x t w x t x t f p x x t dp





         

 For Gaussian wave packet use Gaussian spectral function:

 

2 0

1 4 2

( )

( ) 1 exp

2 p 4 p

p p

f p   

  

   

 

 

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Spreading of Gaussian Wave Packet for Particle at Rest

 The probability density that the particle is located at some location in space is determined from the wave function (in coordinate representation):

( )

*

( ) ( )

 x   x  x   x x 

 For a free particle the Schrödinger equation has the form of a diffusion equation with diffusion coefficient V ( ) 0 x  .

/2 i  m

As time progresses the width of the distribution increases . This width is proportional to the standard deviation of the position x. In quantum mechanics the standard deviation of the probability distribution of a variable is often called the uncertainty in the

variable. If we have an object of mass 1 kg then the time scale for the uncertainty in the position of this object to increase by about is estimated to be years. Hence classical physics is amply adequate to describe the dynamics of macroscopic objects.

2 2 i

m t

  





 ^{t m} ^/ 

^{1/ 2}

 

1 0 m

^6

3 10 

⁶

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Moving Gaussian Wave Packet

( , )

( , ) x t f p ( )

_p

( x x t dp

0

, ) M x t e ( , )

ⁱ

^

^{x t}





     

2

*

2

( ( ) ) ( , ) ( , ) ( , ) 1 exp

2 ( ) 2

x x t x t x t x t

 t

 

  

     

 

 

2

0 0

1 4 2

( )

amplitude function : ( , ) 1 exp

2

p

4

x

x x v t M x t

  

   

   

 

group velocity:

0

p v  m

2 2 2

2

2 2 2

localization in space: 1 4 4

p x

p

t m

 



 

       



2 2

0

0 2 0 0 0 0 0

arctan 2

phase : ( , ) 1 ( ) ( )

2 2 2

p p

x

t p

t m

x t p x x v t x x v t v t

m

 

 

 

         

 

 



 

*

0 0

ˆ ( , ) ( , )

x x t x x t x v t



     

Physical (i.e., measurable) properties are contained in:

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Gaussian Wave Packet: Uncertainty Relations

  ²   ² ^* ²

ˆ ˆ ˆ ˆ ˆ

var( ) x x x ^ ( , ) x t x x ( , ) x t dx  _x



       

  ² ⁰ ² ^* ²

ˆ ˆ ˆ

var( ) p p p ( , ) x t p ( , ) x t dx _p

i x 





  

       ^      

ˆ ˆ var( ) var( ) ˆ ˆ ( )

x p 2

x p x p  t 

     

2

( ,0) 0

0 0

1 4 2

( )

( ,0) ( ,0) 1 exp exp ( )

(2 ) 4 0

ˆ ˆ ( 0)

2 2

i x

x x

x p p

p

x x i

x M x e p x x

t

x p t



  

  



          

      

  

  

     

 



 

← minimum uncertainty state

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Time-Dependent Schrödinger Equation:

Direct Solution

2

ˆ , ˆ

2

( )

i H H 2 V

t m

      

  r



( ) V r

1

ˆ

n n

i t

n



 H

    



 t

2

1 1

2 2

2

n n n

n

k k k

x

k

x

 

    

 

  

 x

 The time dependent Schr ödinger equation can be compactly written a ö

where we assume Hamiltonian for a single particle in a potential .

 Lets try to use naïve Euler method to convert this into a difference equation:

Where the superscript represent time and is the time increment.

The second derivative in the Hamiltonian is approximated by:

where is the interval in x , with similar expressions for the contributions from the y

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Stability Problem of Direct Euler Approach

 This finite difference equation is unstable:

 This is a linear second order difference equation in two variables. We can solve it by Fourier analysis in the x direction. What this means is that x - dependence can be taken to be a superposition of solutions of form :

 Thus, at each time step is multiplied by the amplification factor:

 

1

2

1

,

2

n n n n n

k k k k k

iQ Q t

m x



 

          





e

ikx

     

 

1

exp exp 2 exp

a constant

1 2 1 cos( )

n n

k n n n

n n

i k x

iQ i x i x

x k x

iQ x

 

    

  



                     

 

     



 

²

 

²

1 2 1 cos 1 4 1 cos

1 for 0

iQ x Q x

Q

   

 

   

            

   

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Implicit Method Cure?

 The instability of the Euler method is cured by time-reversal, i.e., we can use the implicit method. For the free particle:

so that “amplification” factor is now:

 However, we now have new problem with unitarity: The implicit method does not preserve unitarity, i.e., it does not keep the normalization integral constant

 

1 1 1 1

1 2 1 , 2

2 n n n n n

k k k k k

iQ Q t

m x

   

 

          





 

²

 

²

1 1

1 2 iQ 1 cos x 1 4 Q 1 cos x 1

 

 

   

 

           

( ) ( ) x

*

x dx 1

  



 This is related to the fact that the amplification factor has magnitude less than or

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A Cure for both Stability and Unitarity

 The unitarity problem is cured by using a more accurate method — evaluate the time derivative at the midpoint of the time interval by taking the average of the implicit and explicit Euler methods approximations for the Hamiltonian.

 For free particle such finite difference method yields the equation:

 This is now finally a stable method that preserves unitarity; for a particle in 1 dimension with potential , the finite difference TDSE is:

 If considered as a set of linear equations for the unknowns , we have a tri-diagonal system of linear equations that can be solved much easier than standard LU decomposition + forward-backward substitution. Note, however, that you need to use complex arithmetic.

   

 

1 1 1 1

1

2

1 1

2

1

2 2

1 1 cos 1 1 cos 1

n n n n n n n n

k k k k k k k k

Q Q

i i

iQ x

  



   

   

              

 

    

  

 

    

( )

 V x   

1 1 1 1 1

1 1 1 1

2

2 2

( ), ,

2 4 ( ) 2

n n n n n n n n n n

k k k k k k k k k k k k

k k

iq irV iq irV

Q t t

V V x q r

m x

    

   

                  

 

   





1 1 1

1ⁿ^

,

2ⁿ^

, ,

ⁿ_K^

   

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Test Example for Computational Algorithm Ensuring Unitary Evolution of Wave Packet

 The figure shows the probability distribution at equal time intervals for a

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Solving Time-Independent Schrödinger Equation via Exact Diagonalization

 Separation of variables to solve the partial differential equation:

 The spatial part has boundary conditions that often lead to an eigenvalue problem, i.e., there is a solution for only for a discrete set of values of This is formally similar to oscillations in a linear chain from Project 3.

 If we label the eigenvalues and the corresponding eigenfunctions then the complete solution is:

Computational Methods in Quantum Mechanics