Advanced Quantum Mechanics 2 lecture 5 Symmetries in quantum mechanics Symmetries in quantum mechanics

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  • Advanced Quantum Mechanics 2 lecture 5

    Symmetries in Quantum Mechanics

    Yazid Delenda

    Département des Sciences de la matière Faculté des Sciences - UHLB

    http://theorique05.wordpress.com/f411

    Batna, 14 December 2014

    (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 1 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator is one which when acts on the Hamiltonian it renders it unchanged:

    ÔĤ = Ĥ

    where Ô is a symmetry operator. Consider for example the Hamiltonian:

    Ĥ = p2

    2m = − ~

    2

    2m

    ∂2

    ∂x2

    for a free particle.

    Ĥ(t+ I) = Ĥ(t), so ÔĤ = Ĥ, where Ô here is the time translation symmetry operator

    Ĥ(x+ a) = Ĥ, so T̂aĤ = Ĥ, where T̂a here is the space translation symmetry operator

    Ĥ(−x) = Ĥ(x), so P̂ Ĥ = Ĥ, where P̂ parity operator (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 2 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator is one which when acts on the Hamiltonian it renders it unchanged:

    ÔĤ = Ĥ

    where Ô is a symmetry operator. Consider for example the Hamiltonian:

    Ĥ = p2

    2m = − ~

    2

    2m

    ∂2

    ∂x2

    for a free particle.

    Ĥ(t+ I) = Ĥ(t), so ÔĤ = Ĥ, where Ô here is the time translation symmetry operator

    Ĥ(x+ a) = Ĥ, so T̂aĤ = Ĥ, where T̂a here is the space translation symmetry operator

    Ĥ(−x) = Ĥ(x), so P̂ Ĥ = Ĥ, where P̂ parity operator (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 2 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator is one which when acts on the Hamiltonian it renders it unchanged:

    ÔĤ = Ĥ

    where Ô is a symmetry operator. Consider for example the Hamiltonian:

    Ĥ = p2

    2m = − ~

    2

    2m

    ∂2

    ∂x2

    for a free particle.

    Ĥ(t+ I) = Ĥ(t), so ÔĤ = Ĥ, where Ô here is the time translation symmetry operator

    Ĥ(x+ a) = Ĥ, so T̂aĤ = Ĥ, where T̂a here is the space translation symmetry operator

    Ĥ(−x) = Ĥ(x), so P̂ Ĥ = Ĥ, where P̂ parity operator (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 2 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator is one which when acts on the Hamiltonian it renders it unchanged:

    ÔĤ = Ĥ

    where Ô is a symmetry operator. Consider for example the Hamiltonian:

    Ĥ = p2

    2m = − ~

    2

    2m

    ∂2

    ∂x2

    for a free particle.

    Ĥ(t+ I) = Ĥ(t), so ÔĤ = Ĥ, where Ô here is the time translation symmetry operator

    Ĥ(x+ a) = Ĥ, so T̂aĤ = Ĥ, where T̂a here is the space translation symmetry operator

    Ĥ(−x) = Ĥ(x), so P̂ Ĥ = Ĥ, where P̂ parity operator (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 2 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator is one which when acts on the Hamiltonian it renders it unchanged:

    ÔĤ = Ĥ

    where Ô is a symmetry operator. Consider for example the Hamiltonian:

    Ĥ = p2

    2m = − ~

    2

    2m

    ∂2

    ∂x2

    for a free particle.

    Ĥ(t+ I) = Ĥ(t), so ÔĤ = Ĥ, where Ô here is the time translation symmetry operator

    Ĥ(x+ a) = Ĥ, so T̂aĤ = Ĥ, where T̂a here is the space translation symmetry operator

    Ĥ(−x) = Ĥ(x), so P̂ Ĥ = Ĥ, where P̂ parity operator (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 2 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator is one which when acts on the Hamiltonian it renders it unchanged:

    ÔĤ = Ĥ

    where Ô is a symmetry operator. Consider for example the Hamiltonian:

    Ĥ = p2

    2m = − ~

    2

    2m

    ∂2

    ∂x2

    for a free particle.

    Ĥ(t+ I) = Ĥ(t), so ÔĤ = Ĥ, where Ô here is the time translation symmetry operator

    Ĥ(x+ a) = Ĥ, so T̂aĤ = Ĥ, where T̂a here is the space translation symmetry operator

    Ĥ(−x) = Ĥ(x), so P̂ Ĥ = Ĥ, where P̂ parity operator (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 2 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator commutes with the Hamiltonian:

    [Ô, Ĥ] = 0

    since

    [Ô, Ĥ]ψ(x) = ÔĤψ(x)− ĤÔψ(x) = ĤÔψ(x)− ĤÔψ(x) =

    where we note that ÔĤ = Ĥ implies that ĤÔ = Ĥ.

    Recall that if two operators  and B̂ are diagonal in some representation:

    Â =

     a1 0 · · · 0 0 a2 · · · 0 ...

    ... . . .

    ... 0 0 0 an

     , B̂ = 

    b1 0 · · · 0 0 b2 · · · 0 ...

    ... . . .

    ... 0 0 0 bn

     , (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 3 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator commutes with the Hamiltonian:

    [Ô, Ĥ] = 0

    since

    [Ô, Ĥ]ψ(x) = ÔĤψ(x)− ĤÔψ(x) = ĤÔψ(x)− ĤÔψ(x) =

    where we note that ÔĤ = Ĥ implies that ĤÔ = Ĥ.

    Recall that if two operators  and B̂ are diagonal in some representation:

    Â =

     a1 0 · · · 0 0 a2 · · · 0 ...

    ... . . .

    ... 0 0 0 an

     , B̂ = 

    b1 0 · · · 0 0 b2 · · · 0 ...

    ... . . .

    ... 0 0 0 bn

     , (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 3 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    A symmetry operator commutes with the Hamiltonian:

    [Ô, Ĥ] = 0

    since

    [Ô, Ĥ]ψ(x) = ÔĤψ(x)− ĤÔψ(x) = ĤÔψ(x)− ĤÔψ(x) =

    where we note that ÔĤ = Ĥ implies that ĤÔ = Ĥ.

    Recall that if two operators  and B̂ are diagonal in some representation:

    Â =

     a1 0 · · · 0 0 a2 · · · 0 ...

    ... . . .

    ... 0 0 0 an

     , B̂ = 

    b1 0 · · · 0 0 b2 · · · 0 ...

    ... . . .

    ... 0 0 0 bn

     , (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 3 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    then

    ÂB̂ =

     a1b1 0 · · · 0 0 a2b2 · · · 0 ...

    ... . . .

    ... 0 0 0 anbn

     = B̂Â so if two operators are diagonal in some representations then they commute:

    [Â, B̂] = 0

    If two hermitian operators commute, [Â, B̂] = 0 with † =  and B̂† = B̂ then there exists a basis |n〉 in which:

    Â|n〉an|n〉 B̂|n〉bn|n〉

    which means these operators are diagonal (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 4 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    then

    ÂB̂ =

     a1b1 0 · · · 0 0 a2b2 · · · 0 ...

    ... . . .

    ... 0 0 0 anbn

     = B̂Â so if two operators are diagonal in some representations then they commute:

    [Â, B̂] = 0

    If two hermitian operators commute, [Â, B̂] = 0 with † =  and B̂† = B̂ then there exists a basis |n〉 in which:

    Â|n〉an|n〉 B̂|n〉bn|n〉

    which means these operators are diagonal (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 4 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    then

    ÂB̂ =

     a1b1 0 · · · 0 0 a2b2 · · · 0 ...

    ... . . .

    ... 0 0 0 anbn

     = B̂Â so if two operators are diagonal in some representations then they commute:

    [Â, B̂] = 0

    If two hermitian operators commute, [Â, B̂] = 0 with † =  and B̂† = B̂ then there exists a basis |n〉 in which:

    Â|n〉an|n〉 B̂|n〉bn|n〉

    which means these operators are diagonal (http://theorique05.wordpress.com/f411) Advanced Quantum Mechanics 2 - lecture 5 4 / 33

  • Symmetries in quantum mechanics

    Symmetries in quantum mechanics

    then

    ÂB̂ =

     a1b1 0 · · · 0 0 a2b2 · · · 0 ...

    ... . . .

    ... 0 0 0 anbn

     = B̂Â so if two operators are diagonal in some representations then they commute:

    [Â, B̂] = 0

    If two hermitian operators commute, [Â, B̂] = 0 with † =  and B̂† = B̂ then there exists a basis |n〉 in which:

    Â|n〉an|n〉 B̂|n〉bn|n〉

    which means these operators are diagonal (http://theoriq