Quantum Two

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Many Particle Systems Revisited

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Having figured out to construct and work in direct product spaces, we are ready to renew our study of the state spaces, state vectors, and operators and observables of many particle systems.

The basic principle underlying this process has already been stated in the introduction:

The state vector of a system of particles is an element of the direct product space

formed from the single-particle spaces associated with each particle.

So let’s consider some examples.

4

Having figured out to construct and work in direct product spaces, we are ready to renew our study of the state spaces, state vectors, and operators and observables of many particle systems.

The basic principle underlying this process has already been stated in the introduction:

The state vector of a system of particles is an element of the direct product space

formed from the single-particle spaces associated with each particle.

So let’s consider some examples.

5

Having figured out to construct and work in direct product spaces, we are ready to renew our study of the state spaces, state vectors, and operators and observables of many particle systems.

The basic principle underlying this process has already been stated in the introduction:

The state vector of a system of particles is an element of the direct product space

formed from the single-particle spaces associated with each particle.

So let’s consider some examples.

6

Having figured out to construct and work in direct product spaces, we are ready to renew our study of the state spaces, state vectors, and operators and observables of many particle systems.

The basic principle underlying this process has already been stated in the introduction:

The state vector of a system of particles is an element of the direct product space

formed from the single-particle spaces associated with each particle.

So now that we understand what a direct product space is, let’s consider some examples.

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The state space of spinless particles moving in 1D:

As the simplest example, consider a collection of spinless particles each moving in one-dimension, along the -axis, say (e.g., a set of particles confined to a quantum wire)

For each particle, there is /are different

state space basis vectors & operators of interest:

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The state space of spinless particles moving in 1D:

As the simplest example, consider a collection of spinless particles each moving in one-dimension, along the -axis, say (e.g., a set of particles confined to a quantum wire)

For each particle, there is /are different

state space basis vectors & operators of interest:

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The state space of spinless particles moving in 1D:

As the simplest example, consider a collection of spinless particles each moving in one-dimension, along the -axis, say (e.g., a set of particles confined to a quantum wire)

For each particle, there is /are different

state space basis vectors & operators of interest:

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The state space of spinless particles moving in 1D:

As the simplest example, consider a collection of spinless particles each moving in one-dimension, along the -axis, say (e.g., a set of particles confined to a quantum wire)

For each particle, there is /are different

state space basis vectors & operators of interest:

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The state space of spinless particles moving in 1D:

The combined space of all particles in this system is then the -fold direct product

of the individual single-particle spaces, and so is spanned by the direct product basis vectors formed from the position eigenstates of each particle, i.e., we can construct the many-particle position states

which satisfy

And in terms of which an arbitrary -particle quantum state of the system can be expanded.

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The state space of spinless particles moving in 1D:

The combined space of all particles in this system is then the -fold direct product

of the individual single-particle spaces, and so is spanned by the direct product basis vectors formed from the position eigenstates of each particle, i.e., we can construct the many-particle position states

which satisfy

And in terms of which an arbitrary -particle quantum state of the system can be expanded.

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The state space of spinless particles moving in 1D:

The combined space of all particles in this system is then the -fold direct product

of the individual single-particle spaces, and so is spanned by the direct product basis vectors formed from the position eigenstates of each particle, i.e., we can construct the many-particle position states

which satisfy

and in terms of which an arbitrary -particle quantum state of the system can be expanded.

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The state space of spinless particles moving in 1D:

The combined space of all particles in this system is then the -fold direct product

of the individual single-particle spaces, and so is spanned by the direct product basis vectors formed from the position eigenstates of each particle, i.e., we can construct the many-particle position states

which satisfy

and in terms of which an arbitrary -particle quantum state of the system can be expanded.

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Thus, we write

Thus, the quantum mechanical description in the position representation involves a single wave function

but it is not simply a complex number assigned to each point in space.

Indeed, it is a simultaneous function of the position coordinates of all the particles in the system.

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Thus, we write

Thus, the quantum mechanical description in the position representation involves a single wave function

but it is not simply a complex number assigned to each point in space.

Indeed, it is a simultaneous function of the position coordinates of all the particles in the system.

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Thus, we write

Thus, the quantum mechanical description in the position representation involves a single wave function

but it is not simply a complex number assigned to each point in space.

Indeed, it is a simultaneous function of the position coordinates of all the particles in the system.

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Thus, we write

Thus, the quantum mechanical description in the position representation involves a single wave function

but it is not simply a complex number assigned to each point in space.

Indeed, it is a simultaneous function of the position coordinates of all the particles in the system.

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Thus, we write

Thus, the quantum mechanical description in the position representation involves a single wave function

but it is not simply a complex number assigned to each point in space.

Indeed, it is a simultaneous function of the position coordinates of all the particles in the system.

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Comment: This space is clearly isomorphic to that of a single particle moving in dimensions, but the interpretation is different.

For a single particle in dimensions the quantity ψ(x₁,…,x_{N}) represents the amplitude that a position measurement of the particle will find it located at the point having the associated Cartesian coordinates .

For particles moving in 1D, the quantity ψ(x₁,…,x_{N}) represents the amplitude that a simultaneous position measurement of all the particles will find the first at , the second at , and so on.

In either case, it follows that operators from different factor spaces automatically commute with one another. 21

Comment: This space is clearly isomorphic to that of a single particle moving in dimensions, but the interpretation is different.

For a single particle in dimensions the quantity ψ(x₁,…,x_{N}) represents the amplitude that a position measurement of the particle will find it located at the point having the associated Cartesian coordinates .

For particles moving in 1D, the quantity ψ(x₁,…,x_{N}) represents the amplitude that a simultaneous position measurement of all the particles will find the first at , the second at , and so on.

In either case, it follows that operators from different factor spaces automatically commute with one another. 22

Comment: This space is clearly isomorphic to that of a single particle moving in dimensions, but the interpretation is different.

For a single particle in dimensions the quantity ψ(x₁,…,x_{N}) represents the amplitude that a position measurement of the particle will find it located at the point having the associated Cartesian coordinates .

For particles moving in 1D, the quantity ψ(x₁,…,x_{N}) represents the amplitude that a simultaneous position measurement of all the particles will find the first at , the second at , and so on.

In either case, it follows that operators from different factor spaces automatically commute with one another. 23

Comment: This space is clearly isomorphic to that of a single particle moving in dimensions, but the interpretation is different.

For a single particle in dimensions the quantity ψ(x₁,…,x_{N}) represents the amplitude that a position measurement of the particle will find it located at the point having the associated Cartesian coordinates .

For particles moving in 1D, the quantity ψ(x₁,…,x_{N}) represents the amplitude that a simultaneous position measurement of all the particles will find the first at , the second at , and so on.

In either case, it follows that operators from different factor spaces automatically commute with one another. 24

Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from.

Thus, from the momentum eigenstates

we can construct the direct product basis

of the many particle momentum representation, which obey

and in terms of which

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Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from.

Thus, from the momentum eigenstates

we can construct the direct product basis

of the many particle momentum representation, which obey

and in terms of which

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Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from.

Thus, from the momentum eigenstates

we can construct the direct product basis

of the many particle momentum representation, which obey

and in terms of which

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Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from.

Thus, from the momentum eigenstates

we can construct the direct product basis

of the many particle momentum representation, which obey

and in terms of which

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Comment: There is also no particular reason to work in the many particle position representation. Indeed, in the state space associated with each particle, there are many different choices of basis sets to choose from.

Thus, from the momentum eigenstates

we can construct the direct product basis

of the many particle momentum representation, which obey

and in terms of which we may expand in terms of a momentum wave function

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The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:30

The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:31

The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:32

The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:33

The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:34

The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:35

The state space of spinless particles moving in 3D:

The extension to particles moving in higher dimensions is straightforward. F spinless particles each moving in 3D, we have for each particle a

state space basis vectors & operators of interest:

giving direct product ONB’s, obeying

and in terms of which we may expand an arbitrary -particle state:36

The wave functions are then functions of the position vectors or momenta of all of the particles of the system.

A little reflection shows that the mathematical description of particles moving in 3 dimensions is mathematically equivalent, both classically and quantum mechanically, to a single particle moving in a space of 3 dimensions.

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The wave functions are then functions of the position vectors or momenta of all of the particles of the system.

A little reflection shows that the mathematical description of particles moving in 3 dimensions is mathematically equivalent, both classically and quantum mechanically, to a single particle moving in a space of 3 dimensions.

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The wave functions are then functions of the position vectors or momenta of all of the particles of the system.

A little reflection shows that the mathematical description of particles moving in 3 dimensions is mathematically equivalent, both classically and quantum mechanically, to a single particle moving in a space of 3 dimensions.

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The wave functions are then functions of the position vectors or momenta of all of the particles of the system.

A little reflection shows that the mathematical description of particles moving in 3 dimensions is mathematically equivalent, both classically and quantum mechanically, to a single particle moving in a space of 3 dimensions.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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How about particles with spin? We just follow the rules, and see what happens. For N spin-1/2 particles moving in 3D, for example, we have individual state spaces that now take into account the spin degrees of freedom, i.e.,

which allows for direct product states in the combined space, obeying

and in terms of which we can expand an arbitrary state.

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We write:

Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system:

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We write:

Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system:

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We write:

Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system:

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We write:

Thus, the quantum mechanical description of this system requires the specification of a wave function having components associated with the different possible spin configurations of the system:

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Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom.

In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle

Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret.

In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 52

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom.

In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle.

Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret.

In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 53

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom.

In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle.

Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret.

In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 54

Thus, although it may get notationally more complicated, there is no real impediment to applying the postulates of quantum mechanics to the state spaces and state vectors of systems of many particles, even those with internal (e.g., spin) degrees of freedom.

In this last example, the basic rules governing the construction of the direct product space imply that all operators (spin and spatial) associated with any particle in the system automatically commute with those of any other particle.

Thus, computation of quantities of interest, while it may lead to complicated multi-dimensional integrals, is nonetheless straightforward to understand and interpret.

In the next segment, we briefly consider the typical Hamiltonian and the subsequent evolution of many particle systems, focusing on the surprisingly useful case of a system of non-interacting particles. 55

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