Basic Linear Alegbra for Quantum Mechanics

[Peter Yu]

I do not understand a work example in the book: ‘Quantum Mechanics DeMystified”.

On page 212, part of Example 7-5:

Given: Let { |a> |b> } be an orthonormal two-dimensional basis

Let Operator A be given by:

A = |a>< a |- i| a><b |+ i| b><a |- |b><b |

Then: (The following part I do not understand)

A squared = (|a>< a|- i| a><b |+ i| b><a |- |b><b |) (|a>< a |- i| a><b |+ i| b><a |- |b><b |)

=|a>< a|(|a>< a |) + |a>< a|(- i| a><b |)- i| a><b |( i| b><a |) - i| a><b |(- |b><b |)

+ i| b><a |(|a>< a|) + i| b><a |(- i| a><b |)- |b><b |( i| b><a |)- |b><b |(- |b><b |)

= |a>< a|- i| a><b |+|a>< a|+ i| a><b |+|b><b |+ i| b><a |- i| b><a |+|b><b |

= 2|a>< a| + 2|b><b |

Most grateful if someone could help!

 

[Quantum Defect]

I do not understand a work example in the book: ‘Quantum Mechanics DeMystified”.

On page 212, part of Example 7-5:

Given: Let { |a> |b> } be an orthonormal two-dimensional basis

Let Operator A be given by:

A = |a>< a |- i| a><b |+ i| b><a |- |b><b |

Then: (The following part I do not understand)

A squared = (|a>< a|- i| a><b |+ i| b><a |- |b><b |) (|a>< a |- i| a><b |+ i| b><a |- |b><b |)

=|a>< a|(|a>< a |) + |a>< a|(- i| a><b |)- i| a><b |( i| b><a |) - i| a><b |(- |b><b |)

+ i| b><a |(|a>< a|) + i| b><a |(- i| a><b |)- |b><b |( i| b><a |)- |b><b |(- |b><b |)

= |a>< a|- i| a><b |+|a>< a|+ i| a><b |+|b><b |+ i| b><a |- i| b><a |+|b><b |

= 2|a>< a| + 2|b><b |

Most grateful if someone could help!

Which step is confusing you?

You might try to think about the "bra" and "ket" as vectors that make sense to you, for example 2-D unit vectors

I.e. |a> = column vector (1,0); <a| = row vector (1,0); |b> = column vector (0,1); <b| = row vector (0,1)

Things like <a|a> = 1 = <b|b> -- you can see this easily with the vectors above. Also, <a|b> = <b|a> = 0

 

[Peter Yu]

Thank you for your response.

I am confused by the first step, the every basic Ket - Bra alegbra operation.

For example:

How to do the multiplication of:

|a>< a|(|a>< a |) = ?

|a>< a|( i| b><a |) = ?

 

[Qiao]

I assume i is just √-1? Then:
|a>< a|(|a>< a |) =|a>< a|a>< a | = |a>< a |

|a>< a|( i| b><a |) = i|a>< a| b><a | = 0

i is just a number and numbers be moved to the front, back where ever you find convenient.

 

[Peter Yu]

Hi Qiao,
Many Many Thanks! I got it now!