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					CHM2S1-A       Introduction to Quantum Mechanics
                          Dr R. L. Johnston
                                                         THE UNIVERSITY
                                                         OF BIRMINGHAM
I: Foundations of Quantum Mechanics

1. Classical Mechanics
   1.1 Features of classical mechanics.
   1.2 Some relevant equations in classical mechanics.
   1.3 Example – The 1-Dimensional Harmonic Oscillator
   1.4 Experimental evidence for the breakdown of classical mechanics.
   1.5 The Bohr model of the atom.

2. Wave-Particle Duality
   2.1 Waves behaving as particles.
   2.2 Particles behaving as waves.
   2.3 The De Broglie Relationship.
3. Wavefunctions
   3.1 Definitions.
   3.2 Interpretation of the wavefunction..
   3.3 Normalization of the wavefunction.
   3.4 Quantization of the wavefunction
   3.5 Heisenberg’s Uncertainty Principle.

4. Wave Mechanics
   4.1 Operators and observables.
   4.2 The Schrödinger equation.
   4.3 Particle in a 1-dimensional box.
   4.4 Further examples.
                   Learning Objectives
•   To appreciate the differences between Classical (CM) and Quantum
    Mechanics (QM).
•   To know the failures in CM that led to the development of QM.
•   To know how to interpret the wavefunction and how to normalize it.
•   To appreciate the origins and implications of quantization and the
    uncertainty principle.
•   To understand wave-particle duality and know the relationships
    between momentum, frequency, wavelength and energy for “particles”
    and “waves”.
•   To be able to write down the Schrödinger equation for particles: in a 1-
    D box; in 1- and 2-electron atoms; in 1- and 2-electron molecules.
•   To know the origins and allowed values of atomic quantum numbers
    and how the energies and angular momenta of hydrogen atomic
    orbitals depend on them.
•   To be able to sketch the angular and radial nodal properties of atomic
    orbitals.
•   To appreciate the origins of sheilding and its effect on the ordering of
    orbital energies in many-electron atoms.
•   To use the Aufbau Principle, the Pauli Principle and Hund’s Rule to
    predict the lowest energy electron configuration for many-electron
    atoms.
•   To appreciate how the Born-Oppenheimer approximation can be used
    to separate electronic and nuclear motion in molecules.
•   To understand how molecular orbitals (MOs) can be generated as
    linear combinations of atomic orbitals and the difference between
    bonding and antibonding orbitals.
•   To be able to sketch MOs and their corresponding electron densities.
•   To construct MO diagrams for homonuclear and heteronuclear diatomic
    molecules.
•   To predict the electron configurations for diatomic molecules, calculate
    bond orders and relate these to bond lengths, strengths and vibrational
    frequencies.
                       References
Fundamentals
• P. W. Atkins, J. de Paula, Atkins' Physical Chemistry (7th edn.),
  OUP, Oxford, 2001.
• D. O. Hayward, Quantum Mechanics for Chemists (RSC Tutorial
  Chemistry Texts 14) Royal Society of Chemistry, 2002.
• W. G. Richards and P. R. Scott, Energy Levels in Atoms and
  Molecules (Oxford Chemistry Primers 26) OUP, Oxford, 1994.

More Advanced
• P. W. Atkins and R. S. Friedman, Molecular Quantum
  Mechanics (3rd edn.) OUP, Oxford, 1997.
• P. A. Cox, Introduction to Quantum Theory and Atomic Structure
  (Oxford Chemistry Primers 37) OUP, Oxford, 1996.
               1. Classical Mechanics

• Do the electrons in atoms and molecules obey Newton’s
  classical laws of motion?

• We shall see that the answer to this question is “No”.

• This has led to the development of Quantum Mechanics – we
  will contrast classical and quantum mechanics.
1.1 Features of Classical Mechanics (CM)

1) CM predicts a precise trajectory for a particle.

                                                 velocity v


                          position r = (x,y,z)




• The exact position (r)and velocity (v) (and hence the momentum
  p = mv) of a particle (mass = m) can be known simultaneously
  at each point in time.

• Note: position (r),velocity (v) and momentum (p) are vectors,
  having magnitude and direction  v = (vx,vy,vz).
2) Any type of motion (translation, vibration, rotation) can have any
   value of energy associated with it
       – i.e. there is a continuum of energy states.



3) Particles and waves are distinguishable phenomena, with different,
   characteristic properties and behaviour.

                Property                 Behaviour

                mass                     momentum
   Particles    position                collisions
                velocity

   Waves        wavelength              diffraction
                frequency                interference
1.2 Revision of Some Relevant Equations in CM

  Total energy of particle:

         E = Kinetic Energy (KE) + Potential Energy (PE)

                T - depends on v        V - depends on r

                                        V depends on the system
                                        e.g. positional, electrostatic PE

                          E = ½mv2 + V

                         E = p2/2m + V              (p = mv)


Note:   strictly E, T, V (and r, v, p) are all defined at a particular
        time (t) – E(t) etc..
• Consider a 1-dimensional system (straight line translational
  motion of a particle under the influence of a potential acting
  parallel to the direction of motion):

• Define:        position          r=x
                 velocity          v = dx/dt
                 momentum          p = mv = m(dx/dt)

                 PE                V
                 force             F = (dV/dx)

• Newton’s 2nd Law of Motion

                    F = ma = m(dv/dt) = m(d2x/dt2)
                         acceleration

• Therefore, if we know the forces acting on a particle we can
  solve a differential equation to determine it’s trajectory {x(t),p(t)}.
1.3 Example – The 1-Dimensional Harmonic Oscillator

                                    x=0
                                          F          NB – assuming no friction or
                               k                     other forces act on the particle
                                              m      (except F).



                                      x

• The particle experiences a restoring force (F) proportional to its
  displacement (x) from its equilibrium position (x=0).

• Hooke’s Law             F = kx

   k is the stiffness of the spring (or stretching force constant of the
   bond if considering molecular vibrations)
                                                                       k

• Substituting F into Newton’s 2nd Law we get:

                 m(d2x/dt2) = kx             a (second order) differential
                                              equation
Solution:
        position           x(t) = Asin(t)              ω          k
                                                                    m
        of particle

                                                    1      k
        frequency                      = /2 =
        (of oscillation)                           2π      m



Note:   frequency depends only on characteristics of the system
        (m,k) – not the amplitude (A)!
                  x                          time period  = 1/ 
                                  
             +A


                                                             t

             A
• Assuming that the potential energy V = 0 at x = 0, it can be
  shown that the total energy of the harmonic oscillator is given
  by:
                            E = ½kA2

• As the amplitude (A) can take any value, this means that the
  energy (E) can also take any value – i.e. energy is continuous.

• At any time (t), the position {x(t)} and velocity {v(t)} can be
  determined exactly – i.e. the particle trajectory can be specified
  precisely.

• We shall see that these ideas of classical mechanics fail when
  we go to the atomic regime (where E and m are very small) –
  then we need to consider Quantum Mechanics.

• CM also fails when velocity is very large (as v  c), due to
  relativistic effects.
1.4 Experimental Evidence for the Breakdown of Classical Mechanics



 • By the early 20th century, there were a number of experimental
   results and phenomena that could not be explained by classical
   mechanics.

 a) Black Body Radiation (Planck 1900)
                                          “UV Catastrophe”
                                            Classical Mechanics
        Energy               2000 K         (Rayleigh-Jeans)
        Radiated

                                1750 K


                                   1250 K
                   0        2000         4000         6000    l/nm
           Planck’s Quantum Theory
• Planck (1900) proposed that the light energy emitted by the
  black body is quantized in units of h ( = frequency of light).

                         E = nh         (n = 1, 2, 3, …)

• High frequency light only emitted if thermal energy kT  h.

• h – a quantum of energy.

• Planck’s constant      h ~ 6.6261034 Js

• If h  0 we regain classical mechanics.

• Conclusions:
      • Energy is quantized (not continuous).
      • Energy can only change by well defined amounts.
b) Heat Capacities (Einstein, Debye 1905-06)

• Heat capacity – relates rise in energy of a material with its rise in
  temperature:
                          CV = (dU/dT)V

• Classical physics             CV,m = 3R (for all T).
• Experiment                    CV,m < 3R (CV as T).
• At low T, heat capacity of solids determined by
  vibrations of solid.

• Einstein and Debye adopted Planck’s hypothesis.

• Conclusion: vibrational energy in solids is quantized:
   – vibrational frequencies of solids can
     only have certain values ()
   – vibrational energy can only change
     by integer multiples of h.
c) Photoelectric Effect (Einstein 1905)

             h                          Photoelectrons ejected with
                                         kinetic energy:
                                  e e ht l cos
                                     -
                                         e t
                                       Po er n
                                                  Ek = h - F

                                               Metal surface
                                               work function = F


•   Ideas of Planck applied to electromagnetic radiation.
•   No electrons are ejected (regardless of light intensity) unless 
    exceeds a threshold value characteristic of the metal.
•   Ek independent of light intensity but linearly dependent on .
•   Even if light intensity is low, electrons are ejected if  is above the
    threshold. (Number of electrons ejected increases with light
    intensity).

•   Conclusion:           Light consists of discrete packets (quanta) of
                          energy = photons (Lewis, 1922).
     d) Atomic and Molecular Spectroscopy
 •     It was found that atoms and molecules absorb and emit light only at
       specific discrete frequencies  spectral lines (not continuously!).
 •     e.g. Hydrogen atom emission spectrum (Balmer 1885)




n1 = 1  Lyman
n1 = 2  Balmer
n1 = 3  Paschen
n1 = 4  Brackett
n1 = 5  Pfund
                                                ν 1      1   1 
                                             ν    RH  2  2 
                                                c λ     n   n2 
                                                         1     
 •     Empirical fit to spectral lines (Rydberg-Ritz): n1, n2 (> n1) = integers.
 •     Rydberg constant RH = 109,737.3 cm-1 (but can also be expressed
       in energy or frequency units).
Revision: Electromagnetic Radiation




A – Amplitude          l – wavelength
 - frequency          c =  x l or  = c / l

       wavenumber  =  / c = 1 / l

c (velocity of light in vacuum) = 2.9979 x 108 m s-1.
  1.5 The Bohr Model of the Atom

  • The H-atom emission spectrum was rationalized by Bohr (1913):
     – Energies of H atom are restricted to certain discrete values
       (i.e. electron is restricted to well-defined circular orbits,
       labelled by quantum number n).
     – Energy (light) absorbed in discrete amounts (quanta =
       photons), corresponding to differences between these
       restricted values:
    e                         E = E2  E1 = h
                n2
                                         E2                 E2
           n1                                              h
                       h
     p+
                                         E1                 E1
                            Absorption          Emission
• Conclusion: Spectroscopy provides direct evidence for quantization of
energies (electronic, vibrational, rotational etc.) of atoms and molecules.
Limitations of Bohr Model & Rydberg-Ritz Equation

•   The model only works for hydrogen (and other one electron
    ions) – ignores e-e repulsion.

•   Does not explain fine structure of spectral lines.

•   Note: The Bohr model (assuming circular electron orbits) is
    fundamentally incorrect.
             2. Wave-Particle Duality

•   Remember: Classically, particles and waves are distinct:
    – Particles – characterised by position, mass, velocity.
    – Waves – characterised by wavelength, frequency.

•   By the 1920s, however, it was becoming apparent that
    sometimes matter (classically particles) can behave like waves
    and radiation (classically waves) can behave like particles.
2.1 Waves Behaving as Particles

a) The Photoelectric Effect
   Electromagnetic radiation of frequency  can be thought of as
   being made up of particles (photons), each with energy E = h .

   This is the basis of Photoelectron Spectroscopy (PES).

b) Spectroscopy
   Discrete spectral lines of atoms and molecules correspond to
   the absorption or emission of a photon of energy h , causing
   the atom/molecule to change between energy levels: E = h .

   Many different types of spectroscopy are possible.
c) The Compton Effect (1923)

•   Experiment: A monochromatic beam of X-rays (li) = incident on
    a graphite block.

•   Observation: Some of the X-rays passing through the block are
    found to have longer wavelengths (ls).


                                 Intensity
                   ls
         li        
                                                               l
                                                 li   ls
• Explanation: The scattered X-rays undergo elastic collisions with
  electrons in the graphite.
   – Momentum (and energy) transferred from X-rays to electrons.

• Conclusion: Light (electromagnetic radiation) possesses momentum.

• Momentum of photon            p = h/l

• Energy of photon              E = h = hc/ l                       p=h/ls
                                                         ls
                                                    li
• Applying the laws of conservation                           
  of energy and momentum we get:
                                                                  e
                                                                   p=mev
                                h 
                                m c 1  cos 
           Δλ  λ s  λ i        
                                e 
    2.2 Particles Behaving as Waves

    Electron Diffraction (Davisson and Germer, 1925)

                                        Davisson and Germer showed that
                                        a beam of electrons could be diffracted
                                        from the surface of a nickel crystal.

                                        Diffraction is a wave property – arises
                                        due to interference between scattered
                                        waves.

                                        This forms the basis of electron
                                        diffraction – an analytical technique for
                                        determining the structures of molecules,
                                        solids and surfaces (e.g. LEED).

NB: Other “particles” (e.g. neutrons,
protons, He atoms) can also be
diffracted by crystals.
2.3 The De Broglie Relationship (1924)

• In 1924 (i.e. one year before Davisson and Germer’s
  experiment), De Broglie predicted that all matter has wave-like
  properties.

• A particle, of mass m, travelling at velocity v, has linear
  momentum p = mv.

• By analogy with photons, the associated wavelength of the
  particle (l) is given by:
                                 h  h
                               λ 
                                 p mv
                     3. Wavefunctions
•   A particle trajectory is a classical concept.
•   In Quantum Mechanics, a “particle” (e.g. an electron) does not
    follow a definite trajectory {r(t),p(t)}, but rather it is best described
    as being distributed through space like a wave.

3.1 Definitions

•   Wavefunction () – a wave representing the spatial distribution of a
    “particle”.
•   e.g. electrons in an atom are described by a wavefunction centred
    on the nucleus.
•  is a function of the coordinates defining the position of the
    classical particle:
    – 1-D     (x)
  – 3-D       (x,y,z) = (r) = (r,,) (e.g. atoms)
•  may be time dependent – e.g. (x,y,z,t)
The Importance of 
•  completely defines the system (e.g. electron in an atom or
   molecule).
• If  is known, we can determine any observable property (e.g.
  energy, vibrational frequencies, …) of the system.
• QM provides the tools to determine  computationally, to
  interpret  and to use  to determine properties of the system.
3.2 Interpretation of the Wavefunction

• In QM, a “particle” is distributed in space like a wave.
• We cannot define a position for the particle.
• Instead we define a probability of finding the particle at any point
  in space.

The Born Interpretation (1926)

       “The square of the wavefunction at any point in space is
           proportional to the probability of finding the particle
                             at that point.”

•   Note: the wavefunction () itself has no physical meaning.
1-D System
• If the wavefunction at point x is (x), the probability of finding
  the particle in the infinitesimally small region (dx) between x and
  x+dx is:

                          P(x)  (x)2 dx
                                 probability density


• (x) – the magnitude of  at point x.

Why write 2 instead of 2 ?
• Because  may be imaginary or complex  2 would be
  negative or complex.
• BUT: probability must be real and positive (0  P  1).
• For the general case, where  is complex ( = a + ib) then:
        2 = *       where * is the complex conjugate of .
                         (* = a – ib)         (NB i   1 )
3-D System
• If the wavefunction at r = (x,y,z) is (r), the probability of finding
  the particle in the infinitesimal volume element d (= dxdydz) is:

                           P(r)  (r)2 d


• If (r) is the wavefunction describing
  the spatial distribution of an electron
  in an atom or molecule, then:

            (r)2 = (r) – the electron density at point r
3.3 Normalization of the Wavefunction

• As mentioned above, probability:      P(r)  (r)2 d
• What is the proportionality constant?
• If  is such that the sum of (r)2 at all points in space = 1, then:
                                 P(x) = (x)2 dx              1-D
                                 P(r) = (r)2 d              3-D

• As summing over an infinite number of infinitesimal steps = integration,
  this means:                                      2
                                              Ptotal 1D      ψx  dx  1
                                                               

                                    2                           2
           Ptotal 3D      ψr  dτ     ψx, y, z  dxdydz  1
                                           

• i.e. the probability that the particle is somewhere in space = 1.

• In this case,  is said to be a normalized wavefunction.
How to Normalize the Wavefunction

                                           2
• If  is not normalized, then:
                                     ψr  dτ  A  1


• A corresponding normalized wavefunction (Norm) can be
  defined:
                    ψ Norm r     ψr 
                                  1
                                  A


                                    2
   such that:            ψ Norm r  dτ  1


• The factor (1/A) is known as the normalization constant
  (sometimes represented by N).
3.4 Quantization of the Wavefunction

  The Born interpretation of  places restrictions
  on the form of the wavefunction:

  (a)  must be continuous (no breaks);

  (b) The gradient of  (d/dx) must be
  continuous (no kinks);

  (c)  must have a single value at any point in
  space;

  (d)  must be finite everywhere;

  (e)  cannot be zero everywhere.

 • Other restrictions (boundary conditions) depend on the exact system.
 • These restrictions on  mean that only certain wavefunctions and  only
  certain energies of the system are allowed.

                   Quantization of   Quantization of E
3.5 Heisenberg’s Uncertainty Principle

    “It is impossible to specify simultaneously, with precision, both the momentum
                              and the position of a particle*”

    (*if it is described by Quantum Mechanics)
                                                                      Heisenberg (1927)

                    px.x  h / 4              (or /2, where  = h/2).

          x       – uncertainty in position
          px      – uncertainty in momentum (in the x-direction)

•   If we know the position (x) exactly, we know nothing about momentum (px).
•   If we know the momentum (px) exactly, we know nothing about position (x).
•   i.e. there is no concept of a particle trajectory {x(t),px(t)} in QM (which applies to
    small particles).
•   NB – for macroscopic objects, x and px can be very small when compared
    with x and px  so one can define a trajectory.
•   Much of classical mechanics can be understood in the limit h  0.
How to Understand the Uncertainty Principle


• To localize a wavefunction () in space (i.e. to specify the
  particle’s position accurately, small x) many waves of
  different wavelengths (l) must be superimposed  large px
  (p = h/l).
                           2 ~ 1




• The Uncertainty Principle imposes a fundamental (not
  experimental) limitation on how precisely we can know (or
  determine) various observables.
• Note – to determine a particle’s position accurately requires use
  of short radiation (high momentum) radiation. Photons colliding
  with the particle causes a change of momentum (Compton
  effect)  uncertainty in p.
       The observer perturbs the system.

• Zero-Point Energy (vibrational energy Evib  0, even at T = 0 K)
  is also a consequence of the Uncertainty Principle:
    – If vibration ceases at T = 0, then position and momentum
      both = 0 (violating the UP).

• Note: There is no restriction on the precision in simultaneously
  knowing/measuring the position along a given direction (x) and
  the momentum along another, perpendicular direction (z):
                          pz.x = 0
• Similar uncertainty relationships apply to other pairs of
  observables.

• e.g. the energy (E) and lifetime () of a state:

                                  E.  

(a) This leads to “lifetime broadening” of spectral lines:
     – Short-lived excited states ( well defined, small ) possess
        large uncertainty in the energy (large E) of the state.
      Broad peaks in the spectrum.

(b) Shorter laser pulses (e.g. femtosecond, attosecond lasers) have
    broader energy (therefore wavelength) band widths.
   (1 fs = 1015 s, 1 as = 1018 s)
                 4. Wave Mechanics
•   Recall – the wavefunction () contains all the information we need to
    know about any particular system.
•   How do we determine  and use it to deduce properties of the
    system?

4.1 Operators and Observables

•   If  is the wavefunction representing a system, we can write:

                            ˆ
                            Qψ  Qψ

    where       Q – “observable” property of system (e.g. energy,
                    momentum, dipole moment …)
                ˆ
                Q – operator corresponding to observable Q.
• This is an eigenvalue equation and can be rewritten as:

                           Qψ  Q  ψ
                           ˆ

             operator Q acting on   function  multiplied
             function              by a number Q
             (eigenfunction)        (eigenvalue)

  (Note:  can’t be cancelled).

Examples:      d/dx (eax) = a eax

               d2/dx2 (sin ax) = a2 sin ax
To find  and calculate the properties (observables) of a system:

                                  ˆ
   1. Construct relevant operator Q
                       ˆ
   2. Set up equation Qψ  Qψ
   3. Solve equation  allowed values of  and Q.


Quantum Mechanical Position and Momentum Operators

   1. Operator for position in the x-direction is just multiplication by x
                              ˆ ψ  xψ
                              x

                                                           ˆ x  
                                                                   d
   2. Operator for linear momentum in the x-direction:     p     
                                                                 i  dx
   pxψ  pxψ
   ˆ                    dψ
                             pxψ
                        i dx
   (solve first order differential equation   , px).
Constructing Kinetic and Potential Energy QM Operators
   1. Write down classical expression in terms of position and momentum.
   2. Introduce QM operators for position and momentum.


Examples
1. Kinetic Energy Operator in 1-D     ˆ
                                      Tx
                  px2                         ˆ    ˆ
                                                   px2    2  d2
                                                             
                                                                             
                                                                             
          CM Tx                      QM     Tx      
                  2m                               2m     2m  dx 2
                                                             
                                                                             
                                                                             
2. KE Operator in 3-D      ˆ
                           T
        CM                                                QM                     “del-squared”


   p  2    px 2  py2  pz2       ˆ  ˆ
                                     p2    2  2
                                                     2   2                
                                                                              2 2
T                               T                                            
                                     2m        x 2 y 2 z 2
                                           2m                                  2m
   2m             2m                                                         

                                                       partial derivatives
                                                       operate on (x,y,z)

                             ˆ
3. Potential Energy Operator V (a function of position)
    PE operator corresponds to multiplication by V(x), V(x,y,z) etc.
4.2 The Schrödinger Equation (1926)

• The central equation in Quantum Mechanics.
• Observable = total energy of system.

Schrödinger Equation   ˆ
                       Hψ  Eψ                ˆ
                                              H   Hamiltonian Operator

                                              E Total Energy
  where   ˆ ˆ ˆ
          H  TV        and   E = T + V.


• SE can be set up for any physical system.
              ˆ
• The form of H depends on the system.
• Solve SE   and corresponding E.
Examples


1. Particle Moving in 1-D        (x)

 ˆ    ˆ    ˆ                         2   2ψ 
                                                Vx ψ  Eψ
 Hψ  Tψ  Vψ  Eψ                 
                                     2m  x 2 
                                              
• The form of V(x) depends on the physical situation:
   – Free particle            V(x) = 0 for all x.
   – Harmonic oscillator      V(x) = ½kx2

2. Particle Moving in 3-D        (x,y,z)

• SE         2   2ψ  2ψ  2ψ 
                                   Vx, y, z ψ  Eψ
                          
                  x 2 y 2 z 2 
              2m                 
           2 2
   or         ψ  Vx, y, z ψ  Eψ       Note: The SE is a second order
           2m                               differential equation
4.3 Particle in a I-D Box

System
   – Particle of mass m in 1-D box of length L.
   – Position x = 0L.
   – Particle cannot escape from box as PE V(x)=  for x = 0, L (walls).
   – PE inside box: V(x)= 0 for 0< x < L.


                                                               
1-D Schrödinger Eqn.

   2   2ψ 
              Eψ
                                              PE (V)
                       (V = 0 inside box).
   2m  x 2 
            


                                                   0
                                                       0   x    L
                      2   2ψ 
                                 Eψ
                    
                      2m  x 2 
                               
• This is a second order differential equation – with general
  solutions of the form:
                        = A sin kx + B cos kx

              2ψ 
                     k 2 A sin kx  B cos kx   k 2 ψ
              x 2 
                   



• SE        
               2   2ψ   2 
                              k 2 ψ  Eψ
               2m  x 2   2m 
                                              
                             


                 k 2 2
               E                   (i.e. E depends on k).
                   2m
Restrictions on 

• In principle Schrödinger Eqn. has an infinite number of solutions.

• So far we have general solutions:
    – any value of {A, B, k}  any value of {,E}.

• BUT – due to the Born interpretation of , only certain values of 
  are physically acceptable:
    – outside box (x<0, x>L) V =      impossible for particle
                                        to be outside the box
          2 = 0        = 0 outside box.
    –  must be a continuous function
               Boundary Conditions      = 0 at x = 0
                                         = 0 at x = L.
Effect of Boundary Conditions


1. x = 0                  = A sin kx + B cos kx = B
                                       0           1
                         =0 B=0
                         = A sin kx                     for all x


2. x = L                  = A sin kL = 0


                A=0 ?                     sin kL = 0 ?        
                (or  = 0 for all x)

   sin kL = 0           kL = n               n = 1, 2, 3, …
                                               (n  0, or  = 0 for all x)
Allowed Wavefunctions and Energies


•   k is restricted to a discrete set of values:    k = n/L


•   Allowed wavefunctions:          n = A sin(nx/L)
                                                           nx
                                                             
•   Normalization:         A = (2/L)       ψn     2
                                                     L
                                                       sin  
                                                          L 
                                k 2 2 n 2 π 2 2
•   Allowed energies:      En        
                                 2m     2mL 2


                                  n 2h2
                            En 
                                   8mL 2
Quantum Numbers

•   There is a discrete energy state (En),
    corresponding to a discrete wavefunction
    (n), for each integer value of n.

•   Quantization – occurs due to boundary
    conditions and requirement for  to be
    physically reasonable (Born interpretation).

•   n is a Quantum Number – labels each
    allowed state (n) of the system and
    determines its energy (En).


•   Knowing n, we can calculate n and En.
Properties of the Wavefunction

                                                           nx
                                                              
•   Wavefunctions are standing waves:       ψn      2
                                                     L
                                                       sin   
                                                           L 

•   The first 5 normalized wavefunctions for the particle in the 1-D
    box:




•   Successive functions possess one more half-wave ( they have a
    shorter wavelength).

•   Nodes in the wavefunction – points at which n = 0 (excluding the
    ends which are constrained to be zero).

•   Number of nodes = (n-1)       1  0; 2  1; 3  2 …
Curvature of the Wavefunction

• If y = f(x)           dy/dx = gradient of y (with respect to x).
                        d2y/dx2 = curvature of y.

                                                          2ψ      
• In QM         Kinetic Energy  curvature of      T   2  ..... 
                                                         x        
                                                                   
• Higher curvature  (shorter l)  higher KE

• For the particle in the 1-D box (V=0):


                               2ψn        n2
KE                E n  Tn               
                              x 2         L2
                                          
                      n 2h2
Energies    En 
                      8mL 2

•   En  n2/L2               En as n (more nodes in n)
                              En as L (shorter box)
    n (or L)               curvature of n
                             KE  En


           E                                E
              2            node


                                       L
             1




                      L1                            L2
•   En  n2             energy levels get further apart as n
                  n             E

                                       9h 2
                  3             E3 
                                       8mL 2



                                       4h 2
                  2             E2 
                                       8mL 2

                                        h2
                  1            E1                        h2
                               0    8mL 2        ZPE 
                                                         8mL 2

•   Zero-Point Energy (ZPE) – lowest energy of particle in box:

                                                h2
                  ZPE  E min  E 1 
                                               8mL 2
•   CM Emin = 0
•   QM E = 0 corresponds to  = 0 everywhere (forbidden).
•   If V(x) = V  0, everywhere in box, all energies are shifted by V.




                          V=0                    V0
         2 2                                               E2=E2+V
       n h
En          2
                 V                E2
       8mL
                                                           E1=E1+V
                                                    V
                                   E1
Density Distribution of the Particle in the 1-D Box

•   The probability of finding the particle
    between x and x+dx (in the state                  
    represented by n) is:
    Pn(x) = n(x)2 dx = (n(x))2 dx (n is real)

                           2  nx
                                                     2
        Pn x     2
                     L
                         sin    dx
                              L 
•   Note: probability is not uniform
                                                           2
     – n 0 at walls (x = 0, L) for all n.
          2=

     – n2 = 0 at nodes (where n = 0).
4.4 Further Examples

(a) Particle in a 2-D Square or 3-D Cubic Box
• Similar to 1-D case, but   (x,y) or (x,y,z).
• Solutions are now defined by 2 or 3 quantum numbers
   e.g. [n,m, En,m]; [n,m,l, En,m,l].
• Wavefunctions can be represented as contour plots in 2-D


(b) Harmonic Oscillator
• Similar to particle in 1-D box, but PE V(x) = ½kx2

(c) Electron in an Atom or Molecule
    ˆ
    T    3-D KE operator
    ˆ
    V    PE due to electrostatic interactions between electron and all
         other electrons and nuclei.

				
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