Topic 5

Question 1

What is the Hamiltonian operator?

Answer 1

It is the total energy of the system.

Ĥ = T̂ + V̂

Question 2

What is the eigenvalue equation?

Answer 2

Ôu(x)=λu(x)
where Ô is a linear operator
u(x) is the eigenfunction
λ is the eigenvalue

Question 3

What is the relationship between eigenvalues

Answer 3

We assume that u(x) denote a set of normalised to unity eigenfunctions

The eigenfunctions corresponding to different eigenvalues , are mutually orthonormal meaning that

∫(∞ to−∞)uₙ∗(x)uₘ(x)dx=
1 when n=m
0 when n≠m

The equation equals to the Kronecker delta: δnm = 1 when n=m, and 0 when
n≠m

Question 4

How can you write arbitrary solutions in abstract space?

Answer 4

As a sum of basis functions as you can take any function
u(x) that obeys the same boundary condition as the uₙ(x) can be expressed as
u(x) =∑ₙaₙuₙ(x) where
aₙ=∫(∞ to−∞)uₙ∗(x)u(x)dx=
aₙ – the coefficients of expansion (or probability amplitudes)