Quantum Mechanics(14/2) Taehwang Son

Functions as vectors If we know , it is possible to obtain all observable values.

In order to deal with in more complex problems, we need to introduce linear algebra.

• Wave function → a list of numbers or a vector• Operators → matrices • Operation → the multiplication of the vector by the operator matrix.

¿𝑂>¿∫𝜓∗ (𝑥 )𝑂𝜓 (𝑥 )𝑑𝑥

Quantum Mechanics(14/2) Taehwang Son

Functions as vectors

The function f(x) is approximated by its values at three points, x1 , x2 , and x3 , and is rep-resented as a vector in a three-dimensional space.

We can imagine that the set of possible values of the argument is a list of numbers (x), and the corresponding set of values of the function (f(x)) is another list..

[𝑥1𝑥2𝑥3⋮

]→[ 𝑓 (𝑥1)𝑓 (𝑥2)𝑓 (𝑥3)⋮

] 𝑓 (𝑥 )=[ 𝑓 (𝑥1)𝑓 (𝑥2)𝑓 (𝑥3)⋮

]

Quantum Mechanics(14/2) Taehwang Son

Functions as vectors Dirac bra-ket notation

bra vector ket vectorcf)

inner product

Now, let’s represent a function as an expansion of orthonormal basis set.

We have merely changed the axes, and hence the coordinates in our new representation of the vector have changed(now they are the numbers c1 , c2 , c3…).

𝑓 (𝑥 )=∑𝑛

𝑐𝑛𝜓𝑛 (𝑥 ) |𝑓 (𝑥 )>≡ [𝑐1∗𝑐2∗𝑐3∗⋯ ] ,< 𝑓 (𝑥 )|≡[𝑐1𝑐2𝑐3⋮

]

Quantum Mechanics(14/2) Taehwang Son

Functions as vectors Expansion coefficients

→ Projection

Identity matrix

Ex)

Hilbert space

The vector space formed by a set of orthogonal functionse.g.

Quantum Mechanics(14/2) Taehwang Son

Linear operator An example of operator

→ Eigen value problem!

Bilinear expansion

𝑑𝑖=∑𝑗

𝐴𝑖𝑗𝑐 𝑗

Quantum Mechanics(14/2) Taehwang Son

Linear operator Trace of an operator

When calculating physical parameters, basis is not important.

Hermitian matrix

• If A is Hermitian, eigenvalue is real and eigenvector is orthogonal each other.• All observables are Hermitian, so they are real value.