     # 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 Â 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 Â 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 Â 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 Â† = Â 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 Â† = Â 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 Â† = Â 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 Â† = Â 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 ##### Quantum Mechanics 3: the quantum mechanics of many ... wimb/dictaat_QM3_old.pdfآ  Quantum Mechanics
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