Simple Harmonic Oscillator Simple Harmonic Oscillator Quantum harmonic oscillator Eigenvalues and...

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Transcript of Simple Harmonic Oscillator Simple Harmonic Oscillator Quantum harmonic oscillator Eigenvalues and...

  • Simple Harmonic Oscillator Classical harmonic oscillator

    Linear force acting on a particle (Hooke’s law):

    F = !kx

    F = ma = m d 2 x

    dt 2 = !kx

    " d 2 x

    dt 2 +#

    2 x = 0, # = k / m

    x(t) = x(0)cos(!t) + p(0)

    m! sin(!t)

    p(t) = m dx

    dt = p(0)cos !t( ) " m!x(0)sin(!t)

    From Newton’s law:

    Position and momentum solutions oscillate in time:

  • Simple Harmonic Oscillator Classical harmonic oscillator

    Classical Hamiltonian

    H = T +V

    T = p 2

    2m

    F = ! "V

    "x #V =

    1

    2 kx

    2 = 1

    2 m$

    2 x 2

    V(x)

    x

    H = p 2

    2m + 1

    2 m!

    2 x 2

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Quantum Hamiltonian: replace x and p variables with operators

    H = T +V = p 2

    2m + 1

    2 m!

    2 x 2

    Define a dimensionless operator

    a = m!

    2! x + i

    1

    2m!! p

    Then

    a † =

    m! 2!

    x + i 1

    2m!! p

    "

    #$ %

    &'

    = m! 2!

    x ( i 1

    2m!! p

    Position, momentum operators obey the canonical commutation relation:

    [x, p] = i!

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Commutation relation:

    a,a †!" #$ =

    m% 2!

    x + i 1

    2m%! p

    &

    '( )

    *+ ,

    m% 2!

    x , i 1

    2m%! p

    &

    '( )

    *+ !

    " - -

    #

    $ . .

    = m% 2! [x, x]+

    i

    2! [p, x],

    i

    2! [x, p]+

    1

    2m%! [p, p]

    = i

    2! (,i!) ,

    i

    2! (i!) = 1

    a,a †!" #$ = 1

    a † ,a!" #$ = %1

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Number operator:

    Hence we can rewrite the Hamiltonian in terms of the number operator:

    N = a † a =

    m! 2!

    x "i 1

    2m!! p

    #

    $% &

    '( m! 2!

    x + i 1

    2m!! p

    #

    $% &

    '(

    = 1

    !! p 2

    2m + 1

    2 m! 2x2

    # $%

    & '( " 1

    2

    N = a † a =

    m! 2!

    x " i 1

    2m!! p

    #

    $% &

    '( m! 2!

    x + i 1

    2m!! p

    #

    $% &

    '(

    H = p 2

    2m + 1

    2 m! 2x2 = !! a†a +

    1

    2

    # $%

    & '(

  • Simple Harmonic Oscillator Quantum harmonic oscillator Number operator:

    [N ,H ] = a † a,H!" #$ = !% a

    † a, a

    † a +

    1

    2

    & '(

    ) *+

    !

    " ,

    #

    $ - = 0

    [N ,a] = a † a,a!" #$ = a

    † a,a[ ] + a†,a!" #$a = %a

    [N ,a † ] = a

    † a,a

    †!" #$ = a † a,a

    †!" #$ + a † ,a

    †!" #$a = a †

    N † = a

    † a( )

    = a † a †( ) †

    = a † a = N

    Commutation relations

    [H ,a] = !! N +1 2,a[ ] = !! N ,a[ ] = "!!a

    [H ,a † ] = !! N +1 2,a†#$ %& = !! N ,a

    †#$ %& = !!a †

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Eigenvalues and eigenfunctions The energy eigenfunctions and eigenvalues can be found by analytically solving the TISE. Here we will use operator algebra:

    Energy eigenvalue equation (TISE):

    H = p 2

    2m + 1

    2 m! 2x2 = !! N +

    1

    2

    " #$

    % &' = !! a†a +

    1

    2

    " #$

    % &'

    H ! n = E

    n ! n

    Ha ! n = aH " !#a( ) !

    n = E

    n " !#( )a !n

    Ha † !

    n = a

    † H + !#a†( ) !n = En + !#( )a

    † ! n

    Notice that:

    The parentheses around ψ are standard (Dirac) notation for states that is independent of x or p representation. More on this notation later.

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Eigenvalues and eigenfunctions

    The state is an energy eigenfunction with eigenvalue

    The state is an energy eigenfunction with eigenvalue

    H ! n = E

    n ! n

    Ha ! n = E

    n " !#( )a !n

    Ha † !

    n = E

    n + !#( )a† !n

    Hence a and a+ are called the raising and lowering (ladder) operators since they raise or lower the energy by a definite amount.

    a ! n

    a † !

    n

    E n ! !"( )

    E n + !!( )

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Eigenvalues and eigenfunctions

    Consider the lowest eigenvalue of H (ground state energy):

    H ! n = E

    n ! n

    H ! 0 = E

    0 ! 0

    The lowering operator a cannot lower the energy of this eigenstate any further. Hence,

    a ! 0 = 0

    !!a†( )a "0 = !!N "0 = H # !! 2

    $ %&

    ' () " 0 = E

    0 # !! 2

    $ %&

    ' () " 0 = 0

    E 0 = !!

    2

    Ground state energy:

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Eigenvalues and eigenfunctions

    H ! n = E

    n ! n

    H ! 0 = E

    0 ! 0

    We have seen that the states

    are energy eigenstates with energy .

    Thus starting with the lowest energy E0, the energy eigenvalues are

    E 0 ,E

    0 + !! ,E

    0 + 2!! ,......

    E n = E

    0 + n!! = n +

    1

    2

    " #$

    % &' !!

    E 0 = !!

    2

    a † !

    n

    E n + !!( )

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Eigenvalues and eigenfunctions

    A unique feature of the quantum harmonic oscillator is that the energy eigenvalues are equally spaced:

    E n = n +

    1

    2

    ! "#

    $ %& !'

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    ! 0 (x) = N

    0 e " x2 x

    c

    2

    N 0

    2 =

    m#

    $! , x

    c =

    2!

    m#

    Consider the ground state:

    ! m"

    2! # 0 + !

    2m"

    $# 0

    $x = 0

    Normalized solution:

    The lowering operator a cannot lower the energy of this eigenstate any further. Hence,

    a ! 0 = 0

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    ! 0 (x) = N

    0 e " x2 x

    c

    2

    N 0

    2 =

    m#

    $! , x

    c =

    2!

    m#

    Now we can calculate the higher energy (excited) states:

    ! 1 (x) = N

    1 a †! 0 (x) = N

    1

    x

    x c

    " x c

    2

    # #x

    $

    %& '

    () ! 0 (x)

    ! 2 (x) = N

    2 a †( ) 2

    ! 0 (x) = N

    2

    x

    x c

    " x c

    2

    # #x

    $

    %& '

    ()

    2

    ! 0 (x)

    ! n (x) = N

    n a †( )

    n

    ! 0 (x) = N

    n

    x

    x c

    " x c

    2

    # #x

    $

    %& '

    ()

    n

    ! 0 (x)

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    ! 0 (x) = N

    0 e " x2 x

    c

    2

    N 0

    2 =

    m#

    $! , x

    c =

    2!

    m#

    Now we can calculate the higher energy (excited) states:

    ! n (x) =

    1

    2 n n! H

    n y( )!0 (x)

    Normalized solutions:

    Hn(y): Hermite polynomials

    y = 2x

    x c

  • Simple Harmonic Oscillator First few Hermite polynomials:

    H 0 (x) = 1, H

    1 (x) = x, H

    2 (x) = 4x

    2 ! 2

    There are many properties known about Hermite polynomials. See http://mathworld.wolfram.com/HermitePolynomial.html or your favourite mathematics book of special functions for more.

  • Simple Harmonic Oscillator Quantum harmonic oscillator

    Ground State Expectation values (verify this using the ladder operators, a and a+. See Example 2.5 in the textbook)

    !p 0 = p

    2

    0 " p

    2

    = m#!

    2

    !x 0 = x

    2

    0 " x

    2

    = !

    2m#

    !x 0 !p

    0 = !

    2

    The ground state is a minimum uncertainty state. Recall that such a state must be Gaussian.