Parth G | You've Heard of SPIN - But How Is it Encoded in the Math of Quantum Physics? Parth G @ParthGChannel | Uploaded 2 years ago | Updated 11 hours ago
The concept of Spin is hard, but the mathematics is actually quite simple!
In this video I wanted to take a look at how we build up our mathematical representation (or at least one of them) of quantum mechanical spin. To do this, we'll start by looking at the spin of an electron, and understanding what it is.
In quantum mechanics, spin is the inherent angular momentum a particle / system has. It does not gain this angular momentum by moving along an angular (curved) path or spinning in some way - the particle just behaves as if it has angular momentum! Any extra angular momentum it gains as a result of its motion is added to the spin of the particle. Spin is a particle property, just like charge or mass.
With electrons, which are "spin-(1/2)" particles, we know that a measurement of its spin along a particular direction (e.g. z-direction) will result in us finding the electron in a "spin up" or "spin down" state. What this actually means is that the size of the electron's spin angular momentum is the same in both cases (i.e. same spin speed). But for spin up the electron behaves as if it's rotating counterclockwise around the axis, and for spin down it's clockwise. We just represent these spins with arrows pointing in the direction (up) or against (down) the axis for simplicity.
Any quantum system, like our electron, can be represented by a wave function. This wave function contains all the information we can know about the electron, such as what state it's in and the probability of finding a given spin state when we next make a measurement on it.
If we want to find out any information about a system, we have to make a measurement on it. One such example is trying to find the spin of our electron along the z direction. Another example is trying to find the particle's momentum in a given direction.
Taking a measurement is mathematically represented by a "measurement operator" being applied to the system's wave function. If the system is already in a nice "eigenstate", or a state that is one of the possible measurement results of our measurement, then making the measurement will not change the system state. In addition to this, the eigenvalue equation tells us the actual value we will measure in the experiment - in this case, the size of the spin of the electron.
If the system is not in an eigenstate, then a measurement will cause the wave function to "collapse" into one of the possible measurement results. The probability of the system collapsing into a particular state can be calculated from the wave function as it was before we made the measurement. This also links to the concept of superposition, since any quantum state can be written as some superposition of the measurement results of any measurement.
As we see in this video, a quantum state (such as the spin up state we could find our particle in) can be easily represented with a vector. And measurement operators can be represented by matrices. Then we can use the rules of linear algebra to see how measurement operators can be applied to a quantum system. We can also use the usual rules of matrix transformations to work out measurement operators in other directions (e.g. x- and y-directions).
We also see how the measurement matrices used to represent the spin measurements in x-, y-, and z-directions are very close to the Pauli matrices that crop up often when discussing spin-(1/2) particles. Lastly, we see how to construct bigger vectors and matrices for systems where there are more than two possible measurement results - it's just easiest to start with two-state systems like the spin up and spin down states of an electron.
Thanks so much for watching - please do check out my socials here:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Music Chanel - Parth G's Shenanigans
Merch - https://parth-gs-merch-stand.creator-spring.com/
Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.
A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera (Sony A6400): https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
Gorillapod Tripod: https://amzn.to/3wQ0L2Q
Timestamps:
0:00 - Spin: Conceptually Hard, Mathematically Easy(ish)
2:50 - Measurement Operators (i.e. the Math Showing How to Measure a System)
3:50 - Mathematical Representation of Spin Wave Functions (as Vectors)
5:29 - Representing Measurement Operators as Matrices in Linear Algebra
6:40 - The Wave Function Collapses Depending on Our Chosen Measurement!
8:17 - Quantum Superposition (Blend) of Different States
9:29 - The Pauli Matrices
9:55 - Constructing Bigger Vectors and Matrices
The concept of Spin is hard, but the mathematics is actually quite simple!
In this video I wanted to take a look at how we build up our mathematical representation (or at least one of them) of quantum mechanical spin. To do this, we'll start by looking at the spin of an electron, and understanding what it is.
In quantum mechanics, spin is the inherent angular momentum a particle / system has. It does not gain this angular momentum by moving along an angular (curved) path or spinning in some way - the particle just behaves as if it has angular momentum! Any extra angular momentum it gains as a result of its motion is added to the spin of the particle. Spin is a particle property, just like charge or mass.
With electrons, which are "spin-(1/2)" particles, we know that a measurement of its spin along a particular direction (e.g. z-direction) will result in us finding the electron in a "spin up" or "spin down" state. What this actually means is that the size of the electron's spin angular momentum is the same in both cases (i.e. same spin speed). But for spin up the electron behaves as if it's rotating counterclockwise around the axis, and for spin down it's clockwise. We just represent these spins with arrows pointing in the direction (up) or against (down) the axis for simplicity.
Any quantum system, like our electron, can be represented by a wave function. This wave function contains all the information we can know about the electron, such as what state it's in and the probability of finding a given spin state when we next make a measurement on it.
If we want to find out any information about a system, we have to make a measurement on it. One such example is trying to find the spin of our electron along the z direction. Another example is trying to find the particle's momentum in a given direction.
Taking a measurement is mathematically represented by a "measurement operator" being applied to the system's wave function. If the system is already in a nice "eigenstate", or a state that is one of the possible measurement results of our measurement, then making the measurement will not change the system state. In addition to this, the eigenvalue equation tells us the actual value we will measure in the experiment - in this case, the size of the spin of the electron.
If the system is not in an eigenstate, then a measurement will cause the wave function to "collapse" into one of the possible measurement results. The probability of the system collapsing into a particular state can be calculated from the wave function as it was before we made the measurement. This also links to the concept of superposition, since any quantum state can be written as some superposition of the measurement results of any measurement.
As we see in this video, a quantum state (such as the spin up state we could find our particle in) can be easily represented with a vector. And measurement operators can be represented by matrices. Then we can use the rules of linear algebra to see how measurement operators can be applied to a quantum system. We can also use the usual rules of matrix transformations to work out measurement operators in other directions (e.g. x- and y-directions).
We also see how the measurement matrices used to represent the spin measurements in x-, y-, and z-directions are very close to the Pauli matrices that crop up often when discussing spin-(1/2) particles. Lastly, we see how to construct bigger vectors and matrices for systems where there are more than two possible measurement results - it's just easiest to start with two-state systems like the spin up and spin down states of an electron.
Thanks so much for watching - please do check out my socials here:
Instagram - @parthvlogs
Patreon - patreon.com/parthg
Music Chanel - Parth G's Shenanigans
Merch - https://parth-gs-merch-stand.creator-spring.com/
Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.
A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL
My Camera (Sony A6400): https://amzn.to/2SjZzWq
ND Filter: https://amzn.to/3qoGwHk
Microphone and Stand (Fifine): https://amzn.to/2OwyWvt
Gorillapod Tripod: https://amzn.to/3wQ0L2Q
Timestamps:
0:00 - Spin: Conceptually Hard, Mathematically Easy(ish)
2:50 - Measurement Operators (i.e. the Math Showing How to Measure a System)
3:50 - Mathematical Representation of Spin Wave Functions (as Vectors)
5:29 - Representing Measurement Operators as Matrices in Linear Algebra
6:40 - The Wave Function Collapses Depending on Our Chosen Measurement!
8:17 - Quantum Superposition (Blend) of Different States
9:29 - The Pauli Matrices
9:55 - Constructing Bigger Vectors and Matrices
