Click here to load reader

The variational principle - unios.hr ... The variational principle The variational principle Quantum mechanics 2 - Lecture 5 Igor Luka cevi c UJJS, Dept. of Physics, Osijek November

  • View
    2

  • Download
    0

Embed Size (px)

Text of The variational principle - unios.hr ... The variational principle The variational principle Quantum...

  • The variational principle

    The variational principle Quantum mechanics 2 - Lecture 5

    Igor Lukačević

    UJJS, Dept. of Physics, Osijek

    November 8, 2012

    Igor Lukačević The variational principle

  • The variational principle

    Contents

    1 Theory

    2 The ground state of helium

    3 The linear variational problem

    4 Literature

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Contents

    1 Theory

    2 The ground state of helium

    3 The linear variational problem

    4 Literature

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    What is a problem we would like to solve?

    To find approximate solutions of eigenvalue problem

    Oφ(x) = ωφ(x)

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    What is a problem we would like to solve?

    To find approximate solutions of eigenvalue problem

    Oφ(x) = ωφ(x)

    A question

    Can you remember any eigenvalue problems?

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    What is a problem we would like to solve?

    To find approximate solutions of Oφ(x) = ωφ(x).

    A question

    Can you remember any eigenvalue problems?

    Hψα = Eαψα , α = 0, 1, . . .

    where E0 ≤ E1 ≤ E2 ≤ · · · ≤ Eα ≤ · · · , 〈ψα|ψβ〉 = δαβ

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Theorem - the variational principle

    Given any normalized function ψ̃ (that satisfies the appropriate boundary conditions), then the expectation value of the Hamiltonian represents an upper bound to the exact ground state energy

    〈ψ̃|H|ψ̃〉 ≥ E0 .

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Theorem - the variational principle

    Given any normalized function ψ̃ (that satisfies the appropriate boundary conditions), then the expectation value of the Hamiltonian represents an upper bound to the exact ground state energy

    〈ψ̃|H|ψ̃〉 ≥ E0 .

    A question

    What if ψ̃ is a ground state w.f.?

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Theorem - the variational principle

    Given any normalized function ψ̃ (that satisfies the appropriate boundary conditions), then the expectation value of the Hamiltonian represents an upper bound to the exact ground state energy

    〈ψ̃|H|ψ̃〉 ≥ E0 .

    A question

    What if ψ̃ is a ground state w.f.?

    〈ψ̃|H|ψ̃〉 = E0

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Proof

    ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Proof

    ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1 On the other hand, (unknown) ψ form a complete set ⇒ |ψ̃〉 =

    ∑ α cα|ψα〉

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Proof

    ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1 On the other hand, (unknown) ψα form a complete set ⇒ |ψ̃〉 =

    ∑ α cα|ψα〉

    So,

    〈ψ̃|ψ̃〉 = 〈∑

    β

    cβψβ

    ∣∣∣∑ α

    cαψα 〉

    = ∑ αβ

    c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸ δαβ

    = ∑ α

    |cα|2 = 1

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Proof

    ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1 On the other hand, (unknown) ψα form a complete set ⇒ |ψ̃〉 =

    ∑ α cα|ψα〉

    So,

    〈ψ̃|ψ̃〉 = 〈∑

    β

    cβψβ

    ∣∣∣∑ α

    cαψα 〉

    = ∑ αβ

    c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸ δαβ

    = ∑ α

    |cα|2 = 1

    Now

    〈ψ̃|H|ψ̃〉 = 〈∑

    β

    cβψβ

    ∣∣∣H∣∣∣∑ α

    cαψα 〉

    ︸ ︷︷ ︸∑ α cαH|ψα〉︸ ︷︷ ︸

    Eα|ψα〉

    = ∑ αβ

    c∗βcαEα 〈ψβ |ψα〉︸ ︷︷ ︸ δαβ

    = ∑ α

    Eα|cα|2

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Proof

    ψ̃ are normalized ⇒ 〈ψ̃|ψ̃〉 = 1 On the other hand, (unknown) ψα form a complete set ⇒ |ψ̃〉 =

    ∑ α cα|ψα〉

    So,

    〈ψ̃|ψ̃〉 = 〈∑

    β

    cβψβ

    ∣∣∣∑ α

    cαψα 〉

    = ∑ αβ

    c∗βcα 〈ψβ |ψα〉︸ ︷︷ ︸ δαβ

    = ∑ α

    |cα|2 = 1

    Now

    〈ψ̃|H|ψ̃〉 = 〈∑

    β

    cβψβ

    ∣∣∣H∣∣∣∑ α

    cαψα 〉

    ︸ ︷︷ ︸∑ α cαH|ψα〉︸ ︷︷ ︸

    Eα|ψα〉

    = ∑ αβ

    c∗βcαEα 〈ψβ |ψα〉︸ ︷︷ ︸ δαβ

    = ∑ α

    Eα|cα|2

    But Eα ≥ E0 , ∀α, hence

    〈ψ̃|H|ψ̃〉 ≥ ∑ α

    E0|cα|2 = E0 ∑ α

    |cα|2 = E0

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    a] Find the ground state energy and w.f. of one-dimensional harmonic oscilator:

    H = − ~ 2

    2m ∆ +

    1

    2 mω2x2 .

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    a] Find the ground state energy and w.f. of one-dimensional harmonic oscilator:

    H = − ~ 2

    2m ∆ +

    1

    2 mω2x2 .

    How to do this using the variational principle...

    (i) pick a trial function which somehow resembles the exact ground state w.f.:

    ψ(x) = Ae−αx 2

    α parameter

    A = 4

    √ 2α

    π from normalization condition (do it for HW)

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    a] Find the ground state energy and w.f. of one-dimensional harmonic oscilator:

    H = − ~ 2

    2m ∆ +

    1

    2 mω2x2 .

    How to do this using the variational principle...

    (i) pick a trial function which somehow resembles the exact ground state w.f.:

    ψ(x) = Ae−αx 2

    α parameter

    A = 4

    √ 2α

    π from normalization condition

    (ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    How to do this using the variational principle...

    (i) pick a trial function which somehow resembles the exact ground state w.f.:

    ψ(x) = Ae−αx 2

    α parameter

    A = 4

    √ 2α

    π from normalization condition

    (ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉

    〈T 〉 = ~ 2α

    2m

    〈V 〉 = mω 2

    On how to solve these kind of integrals, see Ref. [5].

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    How to do this using the variational principle...

    (i) pick a trial function which somehow resembles the exact ground state w.f.:

    ψ(x) = Ae−αx 2

    α parameter

    A = 4

    √ 2α

    π from normalization condition

    (ii) calculate 〈H〉 = 〈T 〉+ 〈V 〉

    〈T 〉 = ~ 2α

    2m

    〈V 〉 = mω 2

    On how to solve these kind of integrals, see Ref. [5].

    〈H〉 = ~ 2α

    2m +

    mω2

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    (iii) minimize 〈H〉 wrt parameter α

    d

    dα 〈H〉 = 0 =⇒ α = mω

    2~

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    (iii) minimize 〈H〉 wrt parameter α

    d

    dα 〈H〉 = 0 =⇒ α = mω

    2~

    (iv) insert back into 〈H〉 and ψ(x):

    〈H〉min = 1

    2 ~ω

    ψmin(x) = 4 √

    π~ e−

    mω 2~ x

    2

    Igor Lukačević The variational principle

  • The variational principle

    Theory

    Example: One-dimensional harmonic oscilator

    (iii) minimize 〈H〉 wrt parameter α

    d

    dα 〈H〉 = 0 =⇒ α = mω

    2~

    (iv) insert back into 〈H〉 and ψ(x):

    〈H〉min = 1

    2 ~ω

    ψmin(x) = 4 √

    π~ e−

    mω 2~ x

    2

     exact ground state energy and w.f.

    A question

    Why did we get the exact energy and w.f.?

Search related