quantum mechanics tunneling & harmonic oscillator

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    2 2

    28

    n

    n hEmL

    2 2

    2 2

    8 0

    mEx h

    Particle in a Box (infinite square well potential)

    Particle in a 1 dimensional Box

    L

    xn

    Lxn

    sin2

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    Particle in a 2 dimensional Box

    2 2 2

    2 2 28 0

    mEx y h

    4 x y yxn nx y x y

    n yn xsin sinL L L L

    222

    8 y

    y

    x

    xnn

    Ln

    Ln

    mhE

    yx

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    22

    28 0 mE

    h

    2 2 2 2

    2 2 2 28 0

    mEx y z h

    Particle in a 3 dimensional Box

    8 x y z

    yx zn n n

    x y z x y z

    n yn x n zsin sin sinL L L L L L

    22 22

    8

    x y zyx z

    n n n

    x y z

    nn nhE

    m L L L

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    Particle in a 3 Dimensional Box

    There are three 1st excited states having

    the same energy. They correspond to

    combinations of the quantum numberswhose squares sum to 6.

    That is

    2

    22

    2,1,11,2,11,1,22

    6

    mLEEE

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    Particle in a 3 Dimensional Box

    The 1st five energy levels for a cubic box.

    n2 Degeneracy

    12 None (2,2,2)

    11 3 ?

    9 3 ?

    6 3 (2,1,1; 1,2,1; 1,1,2)

    3 None (1,1,1)

    4E0

    11/3E0

    2E0

    3E0

    E0

    Energy

    E0 is ground state energy

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    Concept of Modern Physics

    by A. Beiser

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    Another phenomenon explained by the particle-wave is tunneling,which occurs when a particle actually passes through a seeminglyimpenetrable barrier. When a particle hits a barrier, it either hasenough energy to break through or it doesn't and bounces back. But

    with a wave, part of it can pass through while part of it is reflected,making it possible for the particle to appear on the other side.

    Quantum mechanical tunneling; penetration of particlesthrough rectangular potential barrier

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    Quantum mechanical tunneling; penetration of particlesthrough rectangular potential barrier

    Potential Barrier: when a particle approaches a regionin which the entrance of the particle is opposed by someforce, then the region is said to form a barrier for theparticle.

    Tunnel effect: when a particle is able to cross a potentialbarrier even when its energy is less than the barrierheight, then this phenomenon is called tunneleffect. It

    is purely quantum mechanical phenomenon, neverrealizable classically.

    The emission of -particles from atomic nuclei is anexample of tunnel effect

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    Quantum mechanical tunneling; penetration of particlesthrough rectangular potential barrier

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    2 L

    ..

    o

    2 21

    12 2

    2 2

    20 o 22 2

    2 2

    33

    2 2

    Proof : Show that Transmission coeff. T e

    If E

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    1 2

    1 2

    n

    must be finite, single vauled and continous everywhere along x- axis

    Applying boundary conditions

    at x=0 andd d

    dx dx

    These condition when applied to eq (iii)

    2 3

    32at x=Ldd

    dx dx

    1 1 2 2

    1 1 2 2

    L L i L2 2 3

    L L i L

    2 2 3

    A B A Bfor x=0

    i A i B A B

    A e B e A efor x=L

    A e B e i A e

    The ratio between the squares of the magnitudes of the transmitted and the incident

    wave amplitude is

    2 22 L3

    2 2 21

    2

    o

    2 2

    | A | 16

    T= e| A | ( 1)

    (V E) 16where and has a magnitude of the order of unity

    E ( 1)

    T determines the probability of a particle penetrating through barrier and can be

    call

    ed the "transmission coefficient"

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    o2 L

    Approximation transmission probability2m(V E)

    T e where =

    This is the expression for the probabilty

    of penetration

    Quantum mechanical tunneling; penetration of particlesthrough rectangular potential barrier

    - Decay: Penetration of - particle (K.E. few MeV) throughNucleus (Vo25MeV)

    Tunnel Diode, Scanning tunneling Microscope (STM)

    o

    o

    As L , T 0

    As high, (V E), T 0

    or V

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    The tunnel effect provides explanations for the following

    phenomena

    The reverse breakdown of semiconductor diodes.

    The electrical breakdown of insulators.

    The field emission of electrons from a cold metallic surface.

    The switching action of tunnel diodes.

    The emission of-particles from a radioactive elements.

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    Example: Calculate transmission probability

    (i) E=1eV, Vo =10eV, L=0.050nm

    (ii) E=2eV, Vo =10eV, L=0.050nm

    If Length Is Doubled

    o2 L

    16 7

    2m(V E)T e where =

    T e 1.1 10

    7

    T 2.4 10 (Two Times)

    14

    14

    E 1eV T 1.3 10

    E = 2eV T 5.1 10 ( four times)

    Thus, T is more sensitive to the width of thebarrier than to the particle energy here.

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    Infinite square well Finite square well

    Comparison between infinite and finite potential well: Wavefunction

    o2 L

    Approximation transmission probability

    2m(V E)T e where =

    This is the expression for the probabilty

    of penetration

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    Linear Harmonic Oscillator(Parabolic Potential Well)

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    Imp: Linear Harmonic oscillator (Parabolic Potential Well)

    A simple harmonic oscillator is a particle performing 1-D motion underrestoring force (F=-kx) and potential energy V=1/2kx2, where k is force

    constant.

    ** V x2

    The plot against x is a parabola and we may describe the particle as beingin a parabolic potential well.If E Total energy of the particle.Classically : * E can have any value.

    * Particle oscillate back and forth between x=-A and x=+A

    * The velocity of the particle is maximum at the centre of pathand drops to zero at the ends ( x=A)

    * E=K.E.+ P.E.* Thus , classically , the probability of finding the particle is

    minimum at the center and maximum at the ends.

    * Particle can not go beyond. x =

    A

    http://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/9e/HarmOsziFunktionen.png
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    2

    2 22

    2 2

    2

    2

    22 2

    2

    2n n

    Hermite Polynomial

    Quantum Mechanical description :

    d 8 m 1(E kx ) 0 (i)

    dx h 2

    V8 mE mk

    Let = , =h

    d( x ) 0 (ii)

    dx

    (x) C H ( ) e where = x

    where n = 0,1,2,3,..

    .

    n

    1 1 kE (n )h where

    2 2 m

    Wavefunction representations for thefirst eight bound eigenstates, n = 0 to 7.The horizontal axis shows the positionx. The graphs are not normalized

    http://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/9e/HarmOsziFunktionen.png
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    http://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gif
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    Some trajectories of a harmonic oscillator according to Newton's laws of classicalmechanics (A-B), and according to the Schrdinger equation of quantum mechanics(C-F). In (A-B), the particle (represented as a ball attached to a spring) oscillates backand forth. In (C-F), some solutions to the Schrdinger Equation are shown, where thehorizontal axis is position, and the vertical axis is the real part (blue) or imaginarypart (red) of the wavefunction. (C,D,E,F).

    http://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Newton%27s_lawshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Hooke%27s_lawhttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Wavefunctionhttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gifhttp://en.wikipedia.org/wiki/Wavefunctionhttp://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Newton%27s_lawshttp://en.wikipedia.org/wiki/Newton%27s_lawshttp://en.wikipedia.org/wiki/Newton%27s_lawshttp://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gif
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    Comparison of Classical and Quantum Probabilities forHarmonic Oscillator : finding the oscillator at different position

    a) For n=0

    probability is opposite at x=0

    The relative probability of finding it in any interval x is just

    the inverse of its average velocity over that interval.

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    Comparison of Classical and Quantum Probabilities forHarmonic Oscillator : finding the oscillator at different position

    The classical probability is strictlycontained between the vertical lines whichrepresent the classical limit.The quantum probabilities do extend into

    the classically forbidden region,exponentially decaying into that region

    a) For n=0

    probability is opposite at x=0

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    Comparison of Classical and Quantum Probabilities forHarmonic Oscillator : finding the oscillator at different position

    The fact that the overallpicture of probability offinding the oscillator at agiven value ofx convergesfor the quantum andclassical pictures is calledthe

    correspondenceprinciple.

    The greater the quantum number,the closer the quantum physicsapproaches classical physics

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    The energy of the oscillator is E=(p2/2m)+(kx2/2), where p is itsmomentum, displacement from the equilibrium position is x and springconstant is k. In classical physics the minimum energy of the oscillator isEmin=0. Use the uncertainty principle to find an expression for E in terms

    of x only and show that the minimum energy is actually

    minE ( / 2) k / m(or hv/2, where v is frequency of the oscillator).

    Q.7Tut 7

    2 22 2

    2

    min

    2

    3

    24 2

    p x p= hence2 2x

    p 1 kE= kx x (1)

    2m 2 8mx 2

    for E

    dE 0dx

    ( 2)kx 0

    8m x

    ( / 2) ( / 2)x or x

    mk km

    2

    2

    min

    min

    min

    Substituting x in equation 1, we get

    k ( / 2)E

    ( / 2) 2 km8m( )km

    E ( / 2) k / m1 k

    If frequency =2 m

    hE

    2

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    Find the expectation value of for the first two states ofharmonic oscillators.

    Q.6Tut 9

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    Summary : Quantum Mechanics

    1. De-Broglie wavelength

    2. Matter waves: Group velocity and Phase velocity3. Uncertainty principle : Numerical and Applications

    i. Single slit diffraction

    ii. Non-exsistance of electron inside the nucleus.

    4. Wave function: Normalization, average values.5. Schrdinger Equation:

    i. Time dependent

    ii. Time independent

    a) Infinite potential well: Particle in a box

    b) Potential Barrier: Quantum tunnelingc) Harmonic Oscillator