Quantum Mechanics (Unit 1)

chapters 1-2, machinery and operators

For the stern-gerlach experiment - what do we expect due to the the thermal environment of the oven?

a random distribution of spin directions

For the stern-gerlach experiment: how can we determine x-comp of spin based on knowing the z-comp

we cannot know both at same time - knowing one gives us no predictive power

In experiment 4 for SG - what happens when you carefully recombine the beams?

you avoid the QM disturbance and the atoms “remember” their original measurement

Ehrenfest’s theorem: ____

average outcome of QM experiment matches classical prediction

Intrinsic spin is _____

the inherent angular momentum property of elementary particles

particles do not actually “spin”

What is spin quantization? How is it different than classical? what are the values limited between?

the discrete values that spin can take in QM

in classical we could see a continous distribution

limited set of values between S and -S

what is superposition?

combination of states - they exist in multiple states simultaneously until measured

Superposition is ____ combo of ____

linear combo of basis states

How many states in a superposition? What happens to a superposition when it goes thru measurment?

It is never actually in multiple states - it is still a single state

once it goes thru measurement, it collapses onto single state and is no longer super

what is a mixed state? provide example

represents statistical mixture of pure states

ex: a beam that contains mixture of atoms with 50% spin up and 50% spin down

what is a quantum state vector?

mathematical object that fully encodes all the probabilistic info about an isolated quantum system

a hilbert space is ____ with what properties?

complex vector space with inner product

every convergent sum of vectors converges to element inside vector space

what determines the dimensionality of the hilbert space?

the specific quantum state

ex: 2 possible results for spin component = 2 dimensions

In matrix form, how to represent bra and ket?

ket as a column vector, bra as a row vector

complex conjugate’s of each other

how to turn ket into bra and vice versa?

flip the direction of the basis states and take complex conjugate of coefficients

Kets are used as ___, which are the axes of the Hilbert space

Any state Ket is _____ of these basis Kets

basis vectors

linear combination

Orthonormality: basis vector dotted with itself =

basis vector dotted with perpendicular basis vector =

1, because they are parallel

0

Normalization: all vectors describing QM system must be____

what is physical interpretation of this?

normalized, meaning the total probability is 1

means that particle must exist somewhere with certainty

how can we show normality using math?

take inner product of state with itself, must equal 1

Inner product is product of _____

how would you reverse the order? < x | y > =

bra and ket, in that order

take the complex conjugate: < y | x >*

How do I isolate a component in a state vector?

take the inner product of that direction with the state vector to isolate the component in that direction

Born rule: how to find probability of finding certain measurement

take the squared amplitude of the inner product between that measurement bra and state vector Ket (called probability amplitude)

how would i find probability of particle being spin up?

squared amplitude of the inner product between spin up bra and state vector Ket

overall phase: what happens to measurement probabilities?

they don’t change if you multiple state by same complex phase

An ensemble is a conceptual collection of ____, all prepared in the _____

many identical copies of quantum system, all prepared in same initial state

What happens in a totally random ensemble?

every outcome is equally likely

For a pure state - this means that every member is _____

what is it described by?

prepared in the same state (can still be a superposition)

a single normalized ket

for a mixed state - there is a ______

what is it described by?

classical randomness over the different kets in the ensemble

a density operator with probabilites

Mixed states are systems where state is not ____, and represented as a ____

not fully known

probabilistic mixture of pure states

why can we not repeat measurements on the same particle in QM?

because a measurement disturbs the system

A quantum measurement M of a state vector yields ___

outcomes m(n) with probabilities P(n)

All measurement operators are ___, which means what about their eigenvalues?

hermitian, which has real eigenvalues

for a spin-1/2 system, what represents spin observables

the pauli matrices, which has eigenvalue of Hbar/2

What is the expectation value? What is it not?

Average of all outcomes weighted by probability

not the most probable result you could expect, that would be the one with highest probability

if 2 measurements are opposite in magnitude, and have 50% probability each, what is the expected value?

zero

Standard deviation quantifies____

spread of measurements about the mean

If there is only 1 eigenvalue/state, what is the standard value?

zero

2 incompatible observables ____

could not be known or measured simultaneously because measurement of 1 ruined the other

Commutator math formula

If equal to zero what happens? What about if not?

[A, B] = AB - BA

equal to zero - they commute

if not they don’t

does order of operations matter?

only matters if they don’t commut

Commuting observables have common___

common sets of eigenstates

Commuting observables are _____, meaning what?

compatible, we can measure one observable without erasing prior knowledge

Does a product of 2 diagonal matrices depend on order?

No because operator is always diagonal in own basis

and 2 diagonal matrices must share common basis and commute with eachother

Uncertainty principle: product of _____ of 2 observables is related to _____

uncertainties (standard deviations), related to commutator of 2 observables

The uncertainty principle sets the _____

minumum QM uncertainty that would arise in any experiment

Can we know all 3 spin components at same time? what does this imply?

No - we can know 1, but not 2 or 3

implies that spin does not really point in a given direction

general spin - s denotes ____ and the number of beams exiting SG is ____

s denotes spin of system

number of beams is 2s + 1

label m =

what is it?

spin component quantum number

the integer or half integer multiplying h

what does an operator do to a ket? And what does it represent

it eats a Ket and spits out a new one

represents a physical observable like position or momentum

Operators act on quantum states to _____

extract info

Operators are represented by a ___ with what type of eigenvalues

matrix

real eigenvalues

what is the adjoint operator - what does it represent?

the hermitian conjugate of the normal operator

represents same observable since hermitian matrices have real eigenvalues

Adjoints acts in which direction? What do they do to probability?

operates on bras to to the left instead of kets

reverses the action while preserving probabilities

the result of measurement of physical quantity is ____

one of the eigenvalues of the associated observable

What do eigenvalues represent for an observable?

the possible measurement outcomes

For special ket that is not changed by operation of particular operator - what is it?

the ket itself is the eigenvector, and the constant are the eigenvalues

if the system is an eigenvector state - measuring the observable does what?

yields the eigenvalue with uncertainty, collapsing the state onto that value

eigenstates must be mutually _____, and they must span ____

orthogonal

span the whole vector space - everything should be linear combo of it

A diagonal matrix has ___ and what are the eigenvalues

only has diagonal elements

eigenvalues are the diagonal elements

An operator is always ____ in it’s own basis - what are the eigenvectors

always diagonal

eigenvectors are unit vectors in their own basis

what does diagonalization do?

transforms operation into diagonal matrix

For diagonalization process on matrix A
A = PDP^-1

what is each element

D is diagonal matrix with eigenvalues

P is with eigenvectors

P^-1 is inverse

Operator is hermitian if____

how can it act on kets and bras?

it if equals it’s own hermitian adjoint

can act to right on a ket and left on a bra with same result

All observables are ____ with ___eigenvalues

hermitian operators with real eigenvalues

Unitary operators preserves ____

eigenvalues have magnitude of _____

result of the inner product, it doesn’t change the length/angle between them

magnitude of 1 (cannot scale eigenvectors or impact probabilites)

Product of unitary matrix times unitary matrix is ___

still unitary

Unitary - two ways to prove

conjugate transpose of U = U inverse

U dagger times U = 1

outer product is ___ times ____

what does it produce?

results in what type of matrix?

bra times ket

produces another ket

2×2 matrix

A projection operator does what?

eats ket and gives a new ket that is “in the direction” of initial state

projection operator acts on ket and spits out ket that is proportional to ____

what is constant of proportionality?

initial state

constant = overlap between inner product between states

square or projection operator =

what other properties does it have?

square = itself

eigenvalues are either 1 or 0, hermitian

If projection operator eigenvalue = 1, what happens?

If projection operator eigenvalue = 0, what happens?

for 1 - states parallel to initial state (unchanged direction)

for 0 - orthogonal states (annihilated)

How do you calculate projected state

and how to normalize it?

calculate projected state by applying projection operator to input state

divided projected state by length

After obtaining measurement; what happens to state?

system’s state collapses to corresponding eigenstate

S² operator represents what?

what does it commute with?

magnitude of spin vector but gives no direction

other operators like Sz, Sx, Sy

what is eigenvalue and eigenvectors of S²

¾ Hbar²

all states are eigenvectors of it

Can spin vector be fully aligned any axis?

No