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

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Flashcards in 1 - 'Case Studies' of Two-Dimensional Quantum Systems Deck (22)
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1

What is a Stern-Gerlach device?

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.

2

What were the Stern-Gerlach experiments?

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

3

What is Dirac notation?

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.

4

What was the result of the Stern-Gerlach experiments?

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.

5

What are quantized outcomes?

Any measurement of a spin component has quantized outcomes. Meaning that the outcomes of that measurement will be either +/-hbar/2. Quantized spin is a fundamental property of quantum mechanics.

6

What happens in SG experiment 1 - the repeated measurement?

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.

7

What happens in the SG experiment 2 - different measurement?

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.

8

What are kets and bras?

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

9

What happens in the SG experiment 3 - incompatible measurements?

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%.

10

What is angular momentum?

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 = m*v*r*sin(theta) or =I*omega

11

What happens in the SG experiment 4 - interference?

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.

12

How do you express the spin vectors using Dirac notation?

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

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

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

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

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

13

What is the general expression for a spin state?

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

Where a,b=C

14

How do you calculate probabilities?

e.g.

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

15

What is the Born rule?

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

16

What is an observable?

An observable states the condition of the measurement. Namely, it is an operator that describes the quantum state space and the magnetic field orientation.

17

How do you normalise a vector?

Let

|psi'>=c|psi>

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

a*a+b*b=|a|^2+|b|^2

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

18

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

|+>
|->

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

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

19

What is a normalised vector?

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

20

What are the products of common bra-ket combinations?

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.

21

How do you determine the probability of an outcome?

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.

22

What are 'single shot' measurements with probabilistic outcomes?

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