15. States and Superposition Flashcards
In what space are vectors used in to represent state functions?
Hilbert space
What is special about Hilbert space?
It uses vector inner products
See page 7 of document for an inner product representation
Do it
What can be said if the inner product of two functions is equal to 0?
The vectors in Hilbert space are orthogonal
How can a vector be represented in Dirac notation in Hilbert space?
By using a ket | x >
Describe the properties of a Kronecker delta for two components, i and j
It is equal to 1 when i = j
It is equal to 0 otherwise
What type ofequation is the TISE?
A energy eigenvalue equation
State the equation for the eigenvalue spectrum of the harmonic potential
E_n = (n + 1/2) h_bar*ω
What two things can be said if one operator shares a set of eigen functions with another operator?
The two operators commute - their observables are compatible
See page 7 of document for a representation of discrete eigenstates
Do it lol
Define the superposition principle
Where states φ(r) are linear combinations of other functions e.g φ _1 (r), and φ _2 (r)
See page 8 of document for a mathematical representation of the superposition principle
Do it lmao
For the superposition principle, what can be said about the energy of the system if φ _1 (r), and φ _2 (r) are eigenstates of the Hamiltonian with energies E_1 and E_2?
The system φ does not have a well determined energy. If it was to be measured, we would get E_1 OR E_2
- The energy of the system is INDETERMINATE
See page 8 of document for an orthonormality relation
Do it
See page 8 of document for an angular momentum operator example
Do it lol
