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Quantum Spin vs Classical Spin · Acedra
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Quantum Spin vs Classical Spin
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0:02
The quantum spin of fundamental particles such as electrons is profoundly different from the concept of spin in classical physics.
0:13
In classical physics, the spin of an object can be described by an arrow pointing along the axis of rotation.
0:22
The length of the arrow indicates the rate at which the object is spinning.
0:29
An arrow pointed in the opposite direction indicates that the object is spinning in the opposite direction.
0:36
If we change the axis of rotation, the arrow changes accordingly.
0:43
At any given moment in time, this arrow can be thought of as the combination of three arrows pointed in each of the three dimensions.
0:54
We refer to these three arrows as the “component of spin in the X direction”, ... …the “component of spin in the Y direction”, …and the “component of spin in the Z direction.”
1:12
In classical physics, it is possible to know the values of all three arrows simultaneously.
1:21
And in classical physics, these arrows can have any length.
1:31
If the spin of the particle is aligned in the Z direction,
1:36
then in classical physics, the X and Y components of spin would be measured to be zero.
1:47
But, in quantum physics, we never measure zero for the components of an electron’s spin in the X, Y, or Z direction, nor in any other direction.
2:01
It is therefore not possible to accurately represent spin in quantum mechanics with an arrow pointing along an axis of rotation, as is shown here for the case of classical physics.
2:19
This does not accurately describe an electron’s quantum spin, as this picture falsely implies that the X and Y components of spin are zero, which is never the case.
2:34
For an electron, we are only able to measure one of these three components of spin at a time, and there is only one possible length that all three arrows can ever be observed to have.
2:51
For example, the arrow representing the Z component of an electron’s spin is always observed as either being pointed up or pointed down, but the length of this arrow never changes.
3:14
Suppose we measure an electron’s component of spin in the Z direction.
3:27
If the only other measurements we make on this electron are other measurements of spin in the Z direction, then we will always obtain the same outcome as in our original measurement.
3:45
But, the moment we measure the electron’s component of spin in one of the other two directions, we lose all knowledge of its spin in the Z direction.
4:12
After we again measure the electron’s spin in the Z direction, there is a 50% probability that it will be in the opposite direction than it was before.
4:27
If we know the electron’s spin in one direction, then the electron’s spins in the other two directions are in inherently unknowable indeterminate conditions.
5:21
If this electron is entangled with another particle somewhere else in the universe, then it is possible to have a quantum state in which the electron’s spin is inherently unknowable in all directions simultaneously, including directions unaligned with any of these three axes.
5:44
But, if the system consists of just this one electron and nothing else, there will always be exactly one direction in which it is possible to know the spin with certainty, though this direction might or might not be aligned with one of these three axes.
6:04
Let’s focus on systems involving only a single electron, and let’s have the yellow arrow represent the one direction in which it is possible to know the spin with 100% certainty.
6:18
The probabilities of measuring the electron’s spin in all possible directions, including directions not necessarily aligned with one of these three axes, is determined by what we call the quantum spin state of the electron.
6:38
The electron’s quantum spin state is described mathematically by two numbers.
6:49
These can be complex numbers with imaginary components, but let’s initially just consider cases where the imaginary components of both numbers are zero.
7:03
The real components of these two numbers are represented by the two coordinates of this green arrow.
7:15
The red sphere represents the first number, and the blue sphere represents the second number.
7:30
If the first number is “one” and the second number is “zero”, then this indicates that we know with 100% certainty that the Z component of spin is pointed up.
7:54
On the other hand, if the first number is “zero” and the second number is “one”, then this indicates that we know with 100% certainty that the Z component of spin is pointed down.
8:17
This combination of numbers indicates that we know with 100% certainty that the X component of spin is pointed in the negative direction.
8:44
This combination of numbers indicates that we know with 100% certainty that the X component of spin is pointed in the positive direction.
9:09
Let us add this new axis to represent the imaginary component of the second number.
9:48
This combination of numbers indicates that we know with 100% certainty that the Y component of spin is pointed in the positive direction.
10:07
This combination of numbers indicates that we know with 100% certainty that the Y component of spin is pointed in the negative direction.
10:24
The green arrow can be pointed in any direction, so long as it has a length of 1.
10:36
When the electron is not interacting with anything, and we are not making any measurements, the green arrow representing the quantum spin state will never change directions.
10:54
But, the moment we make an observation of one of the components of spin, the direction of the green arrow will change to one of the quantum states where that particular component of spin is known with 100% certainty.
11:23
Let us again focus on all the cases where the imaginary components of both numbers are zero.
11:56
The probability that the Z component of spin will be measured to be pointed up is given by the square of the length of this red line.
12:14
The probability that the Z component of spin will be measured pointed down is given by the square of the length of this blue line.
12:36
In this condition, the square of the length of the red line is 1, and the square of the length of the blue line is zero, indicating a 100% probability that the Z component of spin is pointed up, and a 0% probability that it is pointed down.
13:04
In this condition, with the Z component of spin known with certainty, the squares of the lengths of the orange and purple lines are both equal to one half, indicating that there is a 50% probability that the X component of spin is in the positive or negative direction.
13:38
The probability that the X component of spin will be measured in the positive X direction is given by the square of the length of this orange line.
13:55
And the probability that it will be measured in the negative X direction is given by the square of the length of this purple line.
14:39
In this condition, the square of the length of the orange line is 1, and the square of the length of the purple line is 0, indicating that we know with 100% certainty that the X component of spin is in the positive X direction.
15:01
But, the squares of the lengths of the red and blue lines are now both equal to one half, indicating a 50% probability that the Z component of spin is pointed up or down.
15:24
Let us again add the axis representing the imaginary component of the second number.
15:32
Although the first number can also have an imaginary component, we will just show examples where the imaginary component of the first number is zero, due to the fact that this animation is limited to showing only three spatial dimensions.
15:51
The squares of the lengths of the red and blue lines will continue to indicate the probabilities of measuring the Z component of spin as before.
16:16
If the green arrow is in this position, then we have a 100% probability that the Y component of spin will be measured in the positive Y direction.
16:36
With the green arrow in this orientation, we know the Y component of the spin with 100% certainty, but the squares of the red and blue lines indicate that there is a 50% probability of the Z component of spin being in the positive Z or negative Z direction.
17:01
The more certain we are about the spin of the electron in any one of the three dimensions, the less certain we are about its spin in the other two dimensions.
17:18
When the electron is not interacting with anything, and we are not making any measurements, the green arrow representing the quantum spin state will never change directions.
17:34
But, the moment we make an observation of one of the components of spin, the direction of the green arrow will change to one of the quantum states where that particular component of spin is known with 100% certainty.
17:59
For any given quantum state, we can calculate the probability for the measurements of spin in all three dimensions through the following procedure.
18:12
Every possible quantum spin state for an electron can be written mathematically as follows.
18:25
The “C” variables are constants that can have complex values, with real components and imaginary components.
18:38
We can calculate the probabilities of measuring the Z component of spin through the following equations.
18:52
Every possible quantum spin state can also instead be written as follows.
19:05
As before, the “C” variables are constants that can have complex values.
19:19
We can calculate the probabilities of measuring the X component of spin through the following equations.
19:34
Every possible quantum spin state can also instead be written as follows.
19:47
As before, the “C” variables are constants that can have complex values.
20:00
We can calculate the probabilities of measuring the Y component of spin through the following equations.
20:29
Each of the three different ways shown of writing the quantum state comes from the fact that the spin in each of the three dimensions has a matrix associated with it.
20:45
Each matrix has two eigenvectors.
20:53
Each of these eigenvectors represents one of the quantum states we have been discussing in this video, where the spin in one direction is known with 100% certainty.
21:09
Suppose we want to measure the spin of an electron in a direction that is not aligned with the X, Y, or Z axis, and that we describe this direction with an arrow of length one, which we call a “unit vector.”
21:25
The matrix associated with the spin in this direction is a linear combination of the three matrixes shown previously, in the exact same proportions as the unit vector describing this direction is a linear combination of unit vectors pointed in the X, Y, and Z directions.
21:46
The spin in the direction associated with this new matrix is known with 100% certainty when the quantum state is in one of the eigenvectors for this matrix. The value of the measured spin in this direction is the eigenvalue associated with this eigenvector.
22:06
For a system consisting of just a single electron, every possible quantum spin state corresponds to knowing the spin in some particular direction with 100% certainty. Though, if we have a system with two electrons that are entangled with one another, then it is possible to have a quantum state in which the spins are completely unknown in all direction.
22:33
Much more information is available in the videos titled “Quantum Entanglement”, “Quantum Wave Function Visualization”, “Quantum Operators”, “Schrodinger’s Equation”, and in the other videos on this channel. Please subscribe for notifications when new videos are ready.
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