1 - 'Case Studies' of Two-Dimensional Quantum Systems

Question 1

What is a Stern-Gerlach device?

Answer 1

A device that is used to measure the spin components of atoms. It is made up of an inhomogeneous magnetic field, an atom oven, a collimator, and a detection pad. A produced stream of heated neutral atoms is beamed from the oven through a Collimator, into an inhomogeneous magnetic field, and onto the detector pad.

Question 2

What were the Stern-Gerlach experiments?

Answer 2

They were measuring the spin component of atoms through a magnetic field typically in the z-direction

Question 3

What is Dirac notation?

Answer 3

Dirac notation is used to describe most everything in quantum mechanics and is made up of bras and let’s. Most notably, it is used to describe quantum fluctuations and spin of particles.

Question 4

What was the result of the Stern-Gerlach experiments?

Answer 4

They found the beam of silver atoms was split in two. One deflected upwards, and another deflected downwards in relation to the direction of the magnetic field gradient.

Question 5

What are quantized outcomes?

Answer 5

Any measurement of a spin component has quantized outcomes. Meaning that the outcomes of that measurement will be either +/- hbar/2.

Question 6

What happens in SG experiment 1 - the repeated measurement?

Answer 6

The beam passes through the first device records an even split in the up and down spins. However, the spin-up beam passing through another device will only record a beam in the spin-up direction. As it has already split apart.

Question 7

What happens in the SG experiment 2 - different measurement?

Answer 7

This experiment records an expected even split in the beams through the first device. Through the second device the beam is split evenly again as it is being measured using a different magnetic field gradient. However, the second measurement only affects the atoms from one beam of the first measurement. Meaning 50% of the original stream of atoms.

Question 8

What are kets and bras?

Answer 8

Kets is a part of Dirac notation and looks like |+> or |->. Bras are also a part of Dirac notation and look the opposite of kets. i.e.

Question 9

What happens in the SG experiment 3 - incompatible measurements?

Answer 9

Altering the magnetic field orientation resets the measurements of the devices. The first device is measuring the spin component of a given direction (I.e. z) and gives an even split, the next device measures at a different direction (i.e. x) and also records an even split but this time at half of the atoms so it records 25%/25%. The third device measures the same direction as the first device, namely z, but it has been affected by the measuring of the x device and now records an even split 12.5%/12.5%.

Question 10

What is angular momentum?

Answer 10

It is the product of the moment of inertia and the angular velocity. It is basically linear momentum but with an angle ;).

It is expressed as:

L = mvrsin(theta) or =Iomega

Question 11

What happens in the SG experiment 4 - interference?

Answer 11

This is a slight variation of experiment 3. It used three devices, the first and last oriented in the same field direction, the middle device was set on a perpendicular orientation. However, the split beams produced by the middle device both went straight into the third device without measurement. As a result, the experiment recorded the same result as experiment 1.

Question 12

How do you express the spin vectors using Dirac notation?

|+>+|->
|+>-|->
|+>+i|->
|+>-i|->

Answer 12

|+>+|->=(1;1)

|+>-|->=(1;-1)

|+>+i|->=(1;i)

|+>-i|->=(1;-i)

Question 13

What is the general expression for a spin state?

Answer 13

|psi>=a|+>+b|->

Where a,b=C

Question 14

How do you calculate probabilities?

Answer 14

e.g.

P_+=|a|^2
P_-=|b|^2

Question 15

What is the Born rule?

Answer 15

The Born Rule describes the probabilities of getting either result of quantum spin. I.e. For the operator S_z

P_+=|a|^2
P_-=|b|^2

Question 16

What is an observable?

Answer 16

An observable states the condition of the measurement. Namely, it is an operator that describes the quantum state space and the magnetic field orientation. It is the physical quantity that is measured, Sz in this case, is called an observable.

Question 17

How do you normalise a vector?

Answer 17

Let

|psi’>=c|psi>

=|c|^2
=|c|^2(a+|->)
=|c|^2(aa+ba+ab+b*b

aa+bb=|a|^2+|b|^2

=|c|^2*(|a|^2+|b|^2)=1

Question 18

What are the kets corresponding to spin up or down along the three Cartesian axes?

Answer 18

|+>
|->

|+>_x=1/\sqrt(2)[|+>+|->]
|->_x=1/\sqrt(2)[|+>-|->]

|+>_y=1/\sqrt(2)[|+>+i|->]
|->_y=1/\sqrt(2)[|+>-i|->]

Question 19

What is a normalised vector?

Answer 19

When the dot product of a quantum state vector and its corresponding conjugate vector is equal to 1?

Question 20

What are the products of common bra-ket combinations?

Answer 20

If the bra and ket are of the same sign then they will be equal to 1 opposite signs are equal to 0. So same signs are normalisation and the opposite signs are orthogonality.

Question 21

How do you determine the probability of an outcome?

Answer 21

Taking the spin state vector, you find the absolute product of the corresponding positive or negative bra and the state vector, all squared. To quickly find the other probability, you subtract it from 1.

Question 22

What are ‘single shot’ measurements with probabilistic outcomes?

Answer 22

Simply put, we can predict the path that a particle will take by applying probabilities to its various outcomes. For spin-1/2 particles, there are only two dimensions, two outcomes. Which we can determine

Question 23

What are magnetic moments?

Answer 23

The magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. The direction of the magnetic moment points from the south to the north pole of the magnet (inside the magnet). The magnetic field of a magnetic dipole is proportional to its magnetic dipole moment.

Question 24

What is the magnetic field gradient?

Answer 24

A magnetic field gradient is a variation in the magnetic field with respect to position. A one-dimensional magnetic field gradient is a variation with respect to one direction, while a two-dimensional gradient is a variation with respect to two.

It is what is used in measuring spin. The mag gradient is aligned to an axis, i.e. x,y, or z, and the spin is measured.

Question 25

What is the magnetic force magnitude in an SG experiment?

Answer 25

The resulting z-component of the force is;

Fz = d/dz (muu * B)

This force is perpendicular to the direction of motion and deflects the beam in proportion to the component of the magnetic moment in the direction of the magnetic field gradient.

Question 26

What is the first Postulate of Quantum Mechanics?

Answer 26

The state of a quantum mechanical system, including all the information you can know about it, is represented mathematically by a normalised ket |Psi>

Question 27

What is a state preparation device?

Answer 27

It is the first SG device in a system of analyzers.

Question 28

What is the local hidden variable theory?

Answer 28

It is the theory that there may be some other variables, of which we are ignorant, that would allow us to predict the results.

Question 29

What is the link between measurement and disturbance?

Answer 29

It is a fallacy in quantum mechanics that the measurement of a particle implies that said measurement disturbs the particle. This links to the uncertainty principle in that measurement of a component of a particle alter other components of that particle.

Measuring a particle with light gives momentum to the particle. Shorter light wavelengths give better precision but mean a bigger momentum kick so that the momentum of the particle is changed. Similarly, with electron spin, if you measure the spin in one direction, you disturb the spin in another.

Question 30

What is a quantum state vector?

Answer 30

Kats are used for a mathematical description of a quantum mechanical system. These kets are abstract entities that obey many of the rules you know about ordinary spatial vectors. Hence they are called quantum state vectors.

Question 31

What is Hilbert space?

Answer 31

A Hilbert space is a vector space equipped with an inner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.

Quantum state vectors are part of a vector space that we call a Hilbert space. The dimensionality of the Hilbert space is determined by the physics of the system at hand.

Question 32

What is Postulate 2 of Quantum Mechanics?

Answer 32

To every observable in classical mechanics, there corresponds a linear, Hermitian operator in quantum mechanics.

A physical observable is represented mathematically by an operator A that acts on kets.

Question 33

What is Postulate 3 of Quantum Mechanics?

Answer 33

Every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function.

Question 34

What is Postulate 4 of Quantum Mechanics?

Answer 34

The probability of obtaining the value +- hbar / 2 in a measurement of the observable Sz on a system in the state | Psi > is:

P+- = |< +- | Psi >|^2

Where |+-> is the basis ket of Sz corresponding to the result +- hbar / 2.

Question 35

What are superposition states?

Answer 35

A general spin-1/2 state vector |Psi> can be expressed as a combination of the basis kets |+> and |->.

|Psi> = a|+> + b|->

We refer to such a combination of states as a superposition state.

Question 36

What are the double angle trigonometric formulas?

Answer 36

sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x)
cos(2x) = 1 - 2sin^2(x)
cos(2x) = 2cos^2(x) - 1