Quantum Two 1. 2 Motion in 3D as a Direct Product 3.
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Transcript of Quantum Two 1. 2 Motion in 3D as a Direct Product 3.
![Page 1: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/1.jpg)
Quantum Two
1
![Page 2: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/2.jpg)
2
![Page 3: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/3.jpg)
Motion in 3D as a Direct Product
3
![Page 4: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/4.jpg)
In the last topic, we saw how to actually work in direct product spaces, i.e., how to compute inner products, and construct basis vectors, and various operators of interest.
To make these formal definitions more concrete we consider a few examples.
Consider, e.g., our familiar case of a single spinless quantum particle moving in 3 dimensions.
It turns out that this quantum state space can be written as the direct product
of 3 spaces, each of which is describes a quantum particle moving along one of the three orthogonal Cartesian axes.
4
![Page 5: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/5.jpg)
In the last topic, we saw how to actually work in direct product spaces, i.e., how to compute inner products, and construct basis vectors, and various operators of interest.
To make these formal definitions more concrete we consider a few examples.
Consider, e.g., our familiar case of a single spinless quantum particle moving in 3 dimensions.
It turns out that this quantum state space can be written as the direct product
of 3 spaces, each of which is describes a quantum particle moving along one of the three orthogonal Cartesian axes.
5
![Page 6: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/6.jpg)
In the last topic, we saw how to actually work in direct product spaces, i.e., how to compute inner products, and construct basis vectors, and various operators of interest.
To make these formal definitions more concrete we consider a few examples.
Consider, e.g., our familiar case of a single spinless quantum particle moving in 3 dimensions.
It turns out that this quantum state space can be written as the direct product
of 3 spaces, each of which is describes a quantum particle moving along one of the three orthogonal Cartesian axes.
6
![Page 7: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/7.jpg)
In the last topic, we saw how to actually work in direct product spaces, i.e., how to compute inner products, and construct basis vectors, and various operators of interest.
To make these formal definitions more concrete we consider a few examples.
Consider, e.g., our familiar case of a single spinless quantum particle moving in 3 dimensions.
It turns out that this quantum state space can be written as the direct product of 3 spaces,
each of which is describes a quantum particle moving along one of the three orthogonal Cartesian axes.
7
![Page 8: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/8.jpg)
In the last topic, we saw how to actually work in direct product spaces, i.e., how to compute inner products, and construct basis vectors, and various operators of interest.
To make these formal definitions more concrete we consider a few examples.
Consider, e.g., our familiar case of a single spinless quantum particle moving in 3 dimensions.
It turns out that this quantum state space can be written as the direct product of 3 spaces,
each of which is describes a quantum particle moving along one of the three orthogonal Cartesian axes.
8
![Page 9: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/9.jpg)
For a particle moving along each axis, there is /are different
state space basis vectors & operators of interest:
Construct now the direct product space
which according to the rules has an ONB of direct product states
each labeled by 3 Cartesian coordinates of the position vectors of . 9
![Page 10: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/10.jpg)
For a particle moving along each axis, there is /are different
state space basis vectors & operators of interest:
Construct now the direct product space
which according to the rules has an ONB of direct product states
each labeled by 3 Cartesian coordinates of the position vectors of . 10
![Page 11: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/11.jpg)
For a particle moving along each axis, there is /are different
state space basis vectors & operators of interest:
Construct now the direct product space
which according to the rules has an ONB of direct product states
each labeled by 3 Cartesian coordinates of the position vectors of . 11
![Page 12: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/12.jpg)
For a particle moving along each axis, there is /are different
state space basis vectors & operators of interest:
Construct now the direct product space
which according to the rules has an ONB of direct product states
each labeled by 3 Cartesian coordinates of the position vectors of . 12
![Page 13: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/13.jpg)
For a particle moving along each axis, there is /are different
state space basis vectors & operators of interest:
Construct now the direct product space
which according to the rules has an ONB of direct product states
each labeled by 3 Cartesian coordinates of the position vectors of . 13
![Page 14: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/14.jpg)
For a particle moving along each axis, there is /are different
state space basis vectors & operators of interest:
Construct now the direct product space
which according to the rules has an ONB of direct product states
each labeled by 3 Cartesian coordinates of the position vectors of . 14
![Page 15: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/15.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these independent (quantum) mechanical degrees of freedom. 15
![Page 16: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/16.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these independent (quantum) mechanical degrees of freedom. 16
![Page 17: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/17.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis, i.e.,
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these independent (quantum) mechanical degrees of freedom. 17
![Page 18: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/18.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis, i.e.,
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these independent (quantum) mechanical degrees of freedom. 18
![Page 19: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/19.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis, i.e.,
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these independent (quantum) mechanical degrees of freedom. 19
![Page 20: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/20.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis, i.e.,
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these independent (quantum) mechanical degrees of freedom. 20
![Page 21: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/21.jpg)
According to the rules for taking inner products in direct product spaces,
By assumption any state in this space can be expanded in terms of this basis, i.e.,
or in more compact notation
Thus, each state is represented in this basis by a wave function
of these three independent (quantum) mechanical degrees of freedom. 21
![Page 22: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/22.jpg)
Note that by forming the space as a direct product, the individual components of the position operator automatically commute with one another, since they come from different factor spaces.
Indeed, the canonical commutation relations
automatically follow from the rules for extending operators from different factor spaces in to the direct product space.
The action of the individual components of the position operator and momentum operators also follow from the properties of the direct product space, i.e.,
and so on. 22
![Page 23: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/23.jpg)
Note that by forming the space as a direct product, the individual components of the position operator automatically commute with one another, since they come from different factor spaces.
Indeed, the canonical commutation relations
automatically follow from the rules for extending operators from different factor spaces in to the direct product space.
The action of the individual components of the position operator and momentum operators also follow from the properties of the direct product space, i.e.,
and so on. 23
![Page 24: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/24.jpg)
Note that by forming the space as a direct product, the individual components of the position operator automatically commute with one another, since they come from different factor spaces.
Indeed, the canonical commutation relations
automatically follow from the rules for extending operators from different factor spaces into the direct product space.
The action of the individual components of the position operator and momentum operators also follow from the properties of the direct product space, i.e.,
and so on. 24
![Page 25: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/25.jpg)
Note that by forming the space as a direct product, the individual components of the position operator automatically commute with one another, since they come from different factor spaces.
Indeed, the canonical commutation relations
automatically follow from the rules for extending operators from different factor spaces into the direct product space.
The action of the individual components of the position operator and momentum operators also follow from the properties of the direct product space, i.e.,
and so on. 25
![Page 26: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/26.jpg)
Note that by forming the space as a direct product, the individual components of the position operator automatically commute with one another, since they come from different factor spaces.
Indeed, the canonical commutation relations
automatically follow from the rules for extending operators from different factor spaces into the direct product space.
The action of the individual components of the position operator and momentum operators also follow from the properties of the direct product space, i.e.,
and so on. 26
![Page 27: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/27.jpg)
It is left to the viewer to verify that all other properties of the space of a single particle moving in 3 dimensions follow entirely from the properties of the direct product of 3 one-dimensional factor spaces.
27
![Page 28: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/28.jpg)
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State Space of a Spin ½ Particle as a Direct Product Space
29
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Another situation in which the concept of a direct product space becomes valuable is in treating the internal, or spin degrees of freedom of quantum mechanical particles.
It is a well-established experimental fact that the quantum states of most fundamental particles are not completely specified by properties related simply to their motion through space.
In general, each quantum particle possesses an internal structure characterized by a vector observable , the components , of which transform under rotations like the components of angular momentum.
Such a particle is said to possess spin degrees of freedom.
30
![Page 31: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/31.jpg)
Another situation in which the concept of a direct product space becomes valuable is in treating the internal, or spin degrees of freedom of quantum mechanical particles.
It is a well-established experimental fact that the quantum states of most fundamental particles are not completely specified by properties related simply to their motion through space.
In general, each quantum particle possesses an internal structure characterized by a vector observable , the components , of which transform under rotations like the components of angular momentum.
Such a particle is said to possess spin degrees of freedom.
31
![Page 32: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/32.jpg)
Another situation in which the concept of a direct product space becomes valuable is in treating the internal, or spin degrees of freedom of quantum mechanical particles.
It is a well-established experimental fact that the quantum states of most fundamental particles are not completely specified by properties related simply to their motion through space.
In general, each quantum particle possesses an internal structure characterized by a vector observable , the components , of which transform under rotations like the components of angular momentum.
Such a particle is said to possess spin degrees of freedom.
32
![Page 33: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/33.jpg)
Another situation in which the concept of a direct product space becomes valuable is in treating the internal, or spin degrees of freedom of quantum mechanical particles.
It is a well-established experimental fact that the quantum states of most fundamental particles are not completely specified by properties related simply to their motion through space.
In general, each quantum particle possesses an internal structure characterized by a vector observable , the components , of which transform under rotations like the components of angular momentum.
Such a particle is said to possess spin degrees of freedom.
33
![Page 34: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/34.jpg)
For the constituents of atoms (electrons, neutrons, protons) the internal state of each particle can be represented as a superposition of two orthonormal eigenvectors
of the operator with eigenvalues and (in units of ℏ).
A particle whose internal state has is said to be spin up,
A particle whose internal state has , is said to be spin down.
These two orthogonal internal states of the particle span a 2-dimensional state space, referred to as the particle’s spin space.
The spin-degrees of freedom are assumed to be independent of the spatial degrees of freedom of the center of mass, as with, e.g., a classical spinning top. 34
![Page 35: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/35.jpg)
For the constituents of atoms (electrons, neutrons, protons) the internal state of each particle can be represented as a superposition of two orthonormal eigenvectors
of the operator with eigenvalues and (in units of ℏ).
A particle whose internal state has is said to be spin up,
A particle whose internal state has , is said to be spin down.
These two orthogonal internal states of the particle span a 2-dimensional state space, referred to as the particle’s spin space.
The spin-degrees of freedom are assumed to be independent of the spatial degrees of freedom of the center of mass, as with, e.g., a classical spinning top. 35
![Page 36: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/36.jpg)
For the constituents of atoms (electrons, neutrons, protons) the internal state of each particle can be represented as a superposition of two orthonormal eigenvectors
of the operator with eigenvalues and (in units of ℏ).
A particle whose internal state has is said to be spin up,
A particle whose internal state has , is said to be spin down.
These two orthogonal internal states of the particle span a 2-dimensional state space, referred to as the particle’s spin space.
The spin-degrees of freedom are assumed to be independent of the spatial degrees of freedom of the center of mass, as with, e.g., a classical spinning top. 36
![Page 37: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/37.jpg)
For the constituents of atoms (electrons, neutrons, protons) the internal state of each particle can be represented as a superposition of two orthonormal eigenvectors
of the operator with eigenvalues and (in units of ℏ).
A particle whose internal state has is said to be spin up,
A particle whose internal state has , is said to be spin down.
These two orthogonal internal states of the particle span a 2-dimensional state space, referred to as the particle’s spin space.
The spin-degrees of freedom are assumed to be independent of the spatial degrees of freedom of the center of mass, as with, e.g., a classical spinning top. 37
![Page 38: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/38.jpg)
For the constituents of atoms (electrons, neutrons, protons) the internal state of each particle can be represented as a superposition of two orthonormal eigenvectors
of the operator with eigenvalues and (in units of ℏ).
A particle whose internal state has is said to be spin up,
A particle whose internal state has , is said to be spin down.
These two orthogonal internal states of the particle span a 2-dimensional state space, referred to as the particle’s spin space.
The spin-degrees of freedom are assumed to be independent of the spatial degrees of freedom of the center of mass, as with, e.g., a classical spinning top. 38
![Page 39: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/39.jpg)
The main point here, of course, is that the total state space of a spin-1/2 particle can then be represented as a direct product
of the quantum space describing the particle's translational motion (spanned by the position eigenstates , and the 2-dimensional spin space spanned by the spin eigenstates of .
We can therefore construct an ONB of direct product states
which then automatically satisfy the orthonormality and completeness relations
39
![Page 40: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/40.jpg)
The main point here, of course, is that the total state space of a spin-1/2 particle can then be represented as direct product
of the quantum space describing the particle's translational motion (spanned by the position eigenstates , and the 2-dimensional spin space spanned by the spin eigenstates of .
We can therefore construct for the total space, an ONB of direct product states
which then automatically satisfy the orthonormality and completeness relations
40
![Page 41: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/41.jpg)
The main point here, of course, is that the total state space of a spin-1/2 particle can then be represented as direct product
of the quantum space describing the particle's translational motion (spanned by the position eigenstates , and the 2-dimensional spin space spanned by the spin eigenstates of .
We can therefore construct for the total space, an ONB of direct product states
which then automatically satisfy the orthonormality and completeness relations
and
41
![Page 42: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/42.jpg)
The main point here, of course, is that the total state space of a spin-1/2 particle can then be represented as direct product
of the quantum space describing the particle's translational motion (spanned by the position eigenstates , and the 2-dimensional spin space spanned by the spin eigenstates of .
We can therefore construct for the total space, an ONB of direct product states
which then automatically satisfy the orthonormality and completeness relations
and
42
![Page 43: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/43.jpg)
An arbitrary state of a spin-1/2 particle can then be expanded in this basis in the usual way
and thus requires a two component wave function (or spinor).
In other words, to specify the state of the system we must provide two separate complex-valued functions ψ₊(r) and ψ₋(r) such that
gives the probability density to find the particle spin-up at
gives the probability density to find the particle spin-down at
43
![Page 44: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/44.jpg)
An arbitrary state of a spin-1/2 particle can then be expanded in this basis in the usual way
and thus requires a two component wave function (or spinor).
In other words, to specify the state of the system we must provide two separate complex-valued functions ψ₊(r) and ψ₋(r) such that
gives the probability density to find the particle spin-up at
gives the probability density to find the particle spin-down at
44
![Page 45: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/45.jpg)
An arbitrary state of a spin-1/2 particle can then be expanded in this basis in the usual way
and thus requires a two component wave function (or spinor).
In other words, to specify the state of the system we must provide two separate complex-valued functions ψ₊(r) and ψ₋(r) such that
gives the probability density to find the particle spin-up at
gives the probability density to find the particle spin-down at
45
![Page 46: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/46.jpg)
An arbitrary state of a spin-1/2 particle can then be expanded in this basis in the usual way
and thus requires a two component wave function (or spinor).
In other words, to specify the state of the system we must provide two separate complex-valued functions ψ₊(r) and ψ₋(r) such that
gives the probability density to find the particle spin-up at
gives the probability density to find the particle spin-down at
46
![Page 47: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/47.jpg)
An arbitrary state of a spin-1/2 particle can then be expanded in this basis in the usual way
and thus requires a two component wave function (or spinor).
In other words, to specify the state of the system we must provide two separate complex-valued functions ψ₊(r) and ψ₋(r) such that
gives the probability density to find the particle spin-up at
gives the probability density to find the particle spin-down at
47
![Page 48: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/48.jpg)
Note also that, through the rules of the inner product,
all spin operators
automatically commute
with all spatial operators
Thus, the concept of a direct product space arises in many different situations in quantum mechanics.
When such a structure is properly identified it can help elucidate the structure of the underlying state space.
48
![Page 49: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/49.jpg)
Note also that, through the rules of the inner product,
all spin operators
automatically commute
with all spatial operators
Thus, the concept of a direct product space arises in many different situations in quantum mechanics.
When such a structure is properly identified it can help elucidate the structure of the underlying state space.
49
![Page 50: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/50.jpg)
Note also that, through the rules of the inner product,
all spin operators
automatically commute
with all spatial operators
Thus, the concept of a direct product space arises in many different situations in quantum mechanics.
When such a structure is properly identified it can help elucidate the structure of the underlying state space.
50
![Page 51: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/51.jpg)
Note also that, through the rules of the inner product,
all spin operators
automatically commute
with all spatial operators
Thus, the concept of a direct product space arises in many different situations in quantum mechanics.
When such a structure is properly identified it can help elucidate the structure of the underlying state space.
51
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Note also that, through the rules of the inner product,
all spin operators
automatically commute
with all spatial operators
Thus, the concept of a direct product space arises in many different situations in quantum mechanics.
When such a structure is properly identified it can help elucidate the structure of the underlying state space.
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Having explored briefly some concrete examples of direct product spaces, we return in our next segment to the subject which provided the motivation for our mathematical exploration of them, namely
A description of the state space, possible basis vectors, and operators of interest, for a system of N particles of various types (spinless or not) moving in different numbers of possible spatial dimensions.
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Having explored briefly some concrete examples of direct product spaces, we return in our next segment to the subject which provided the motivation for our mathematical exploration of them, namely …
a description of the state space, possible basis vectors, and operators of interest, for a system of N particles of various types (spinless or not) moving in different numbers of possible spatial dimensions.
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![Page 55: Quantum Two 1. 2 Motion in 3D as a Direct Product 3.](https://reader038.fdocuments.in/reader038/viewer/2022110213/5697bfbe1a28abf838ca2d8e/html5/thumbnails/55.jpg)
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