Quantum Mechanics
What are the rules for a hermitian conjugate?
-The Hermitian conjugate of a bra is the corresponding ket, and vice-versa. -The Hermitian conjugate of a scalar (a complex number) is its complex conjugate. -The Hermitian conjugate of the Hermitian conjugate of anything(operators, bras, kets, scalars) is itself - e.g. (A†)† =A -The Hermitian conjugate of any sum of terms is the sum of the Hermitian conjugates -e.g.(Aˆ + Bˆ)† = Aˆ† + Bˆ† -Given any sequence of scalars, bras, kets, inner products, and/or operators, written in bra-ket notation, its Hermitian conjugatecan be computed by reversing the order of the components, and taking the Hermitian conjugate of each.
What is the condition that must be satisfied to be an orthonormal basis?
<ui|uj> = delta ij
When are two vectors orthogonal?
<u|v>=0
How are the basis states of position and momentum related?
A Fourier transform.
How is the matrix of an operator defined?
A Hermitian operator is diagonal in the basis of its eigenstates.
What does it mean to be linearly independent?
As for vectors in a finite-dimensional vector space, a basis for Hilbert space should consists of a set of linearly independent vectors. A set of {|u>, |v>, . . .} is called linearly independent if any linear combination a |u>+b |v>+. . . of the vectors equals zero only if all coefficients (a, b, . . .) equal zero
What does hermitian mean?
A† = A.
How can you improve the variational principle?
By using a variational parameter,this allows you to take a family of guesses
What properties must two vectors have for the inner product to be able to be taken?
Conjugate symmetry in the inner product. <u|v>=<v|u>* Linearity in the second argument (inner product of |a>+|b> is <x|a>+<x|b> Positive definiteness: <u|u> >=0, where zero iff |u>=0
What is the link between Hermitian operators and observables?
Every Hermitian operator corresponds to an observable
A transformation can be represented by the exponent of a generator, what are they.
For spatial and rotational:
What is the link between hermitian operators and an eigen basis?
Given any Hermitian operator, we can find a set of eigenstates for that operator which form an orthonormal basis for all of Hilbert space. We will call such a set of states an eigenbasis.
What is Noether's theorem?
If A generates a symmetry of H then A is a conserved quantity, and likewise, if A is conserved, it generates a symmetry of H .
When is an observable conserved?
If it;s expectation value is time-independent for any initial state:
What can pertubation theory do to degenerate values?
It can lift/reduce the degeneracy
How are states of spin with numbers s and m connected?
States with equal values for the quantum number s of total spin, but with di erent values for the quantum number m labeling the z- component of spin, are connected to each other by raising and lowering operators:
Given an example of incompatible observables.
The Sz and Sx operators cause incompatible observables because if one measurement follows the other it disturbs the first measurement.
What does being a complete space mean?
The limit of any converging sequence of vectors is in the vector space
What is the ground state?
The state with the lowest possible energy
What does the variational principle give?
The upper bound for the ground energy state.
What are the requirements to be a Hilbert space?
The vector space must have an inner product and be complete.
How do you transfer from the |m1,m2> basis to the |s,m> basis?
Using Clebsch-Gordan coefficients such that :
What is the equation for a perturbed hamiltonian?
Where lamda is a small number and H0 is an already known hamiltonian.
What is the norm of a vector?
|| |u> || = sqrt(<u|u>) or:
