Topic 1 Flashcards
What is a (quantum) state?
A quantum state is an abstract vector.
It can be represented by a complex vector of length 1.
It is written as |ψ⟩.
It can be an eigenstate or eigenvector
The state lives in multi-dimensional state space.
What is a quantum mechanical system?
Quantum mechanical system mathematically measures an observable.
It exists in a state which completely describes the system. and it encodes all the information required to predict the results of measurement upon the system.
What is a Hilbert space?
An abstract multi-dimensional state space where the axes correspond to the eigenstates of a given measurement and the coordinates of the end of the vector is the amplitudes of the various eigenstates composing of the vector.
What is the operator?
Measures the value of a particular observable
It can be the measurements on a system represented by mathematical objects
It acts on everything to the right and the order that is is applied matters (NON-CUMULATIVE)
It is denoted as A^(with a hat)
Special examples: Position, differential and momentum
What is an eigenvalue?
When we make a measurement on a randomly chosen quantum state the outcome corresponds to one of the eigenvalues of the operator
Also the value of the observable
What is the eigenvalue equation?
A^|ψ_a⟩=a|ψ_a⟩
|ψ_a⟩ is the eigenstate corresponding to the eigenvalue a
What is Hermitian?
Hermitian have special properties They are real They are complex Orthogonal They can be normalised
What is the Born Rule?
The probability of finding a measurement outcome
P(a)=|⟨ψ_a|ψ⟩|^2
What is the expectation value?
The weighted mean of all probable results
- Each are copies
- All systems are in the same quantum mechanical system but the state is unknown
⟨A⟩=⟨ψ|A^|ψ⟩
What happened after a measurement is taken?
Operating on an arbitrary quantum state vector changes it to become an eigenstate/eigenvector by collapse/projection on the state
If a measurement is taken immediately after the eigenstate of a quantum operator can only be used as a basis for another operator if the operators commute
If operators share eigenstate there is no further collapse if not the system will collapse into a new eigenstate
What are the important operators in QM?
Position operator: x^
Momentum operator: p^_x=-iħ∂/∂x
- Position and Momentum cannot be measured simultaneously with total accuracy
- The commutator is non -zero
- They cannot share the same eigenstate as measuring one after a subsequent measurement of the other causes the state to change due to the collapse/projection
How do you evaluate the commutator of two operators?
Multiply by an arbitrary function(ψ) on the right
[xˆ, pˆ_x]ψ ≡ (ˆxpˆ_x − pˆ_xxˆ)ψ = ˆxpˆ_xψ − pˆ_xxψ, ˆ = x(−iħ∂/∂x)ψ −(−iħ∂/∂x)xψ = −iħx∂ψ/∂x + iħ(∂x/∂xψ + x∂ψ/∂x) = iħψ
What is a collapse?
When you make
measurement the state of the system doesn’t tell you the exact result it could be an infinite result, the eigenvalue gives you one result.
The collapse is basically going from a state of uncertainty to a very definite state associated with the eigenvalue.
What is HUP?
ΔxΔp_x≥ħ/2
It applies to all pairs of observables represented by non-commuting operators
What is a continuous system?
A system with infinite number of eigenstates.
position in cartesian
