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Quantum micro canonical ensemble
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In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (the "external world") that acts as a generator of random phase. The introduction, however, is not quite clear to me:
The wave function $Psi$ of a truly isolated system may be expressed
as a linear superposition of a complete orthonormal set of stationary
wave functions $phi_n$
$$Psi=sum_n c_n phi_n$$
Here we can regard the system plus the external world as a truly
isolated system. The wave function $Psi$ for this whole system will
depend on both the coordinates of the system under consideration and
the coordinates of the external world. If $phi_n$ denotes a
complete set of orthonormal wave functions of the system, then $Psi$
is still formally given by the above equation, but the $c_n$ are to
be interpreted as wave functions of the external world.
Why is that? The definition of $c_n$is
$$c_n= langle phi_n|Psi rangle$$
which is a complex number. How do I get from there to the $c_n$ being wave functions of the outside world?
quantum-mechanics statistical-mechanics waves phase-space
edited 2 hours ago
DanielSank
16.8k44978
asked 5 hours ago
Wasserwaage
1175
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up vote
1
down vote
favorite
In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (the "external world") that acts as a generator of random phase. The introduction, however, is not quite clear to me:
The wave function $Psi$ of a truly isolated system may be expressed
as a linear superposition of a complete orthonormal set of stationary
wave functions $phi_n$
$$Psi=sum_n c_n phi_n$$
Here we can regard the system plus the external world as a truly
isolated system. The wave function $Psi$ for this whole system will
depend on both the coordinates of the system under consideration and
the coordinates of the external world. If $phi_n$ denotes a
complete set of orthonormal wave functions of the system, then $Psi$
is still formally given by the above equation, but the $c_n$ are to
be interpreted as wave functions of the external world.
Why is that? The definition of $c_n$is
$$c_n= langle phi_n|Psi rangle$$
which is a complex number. How do I get from there to the $c_n$ being wave functions of the outside world?
quantum-mechanics statistical-mechanics waves phase-space
edited 2 hours ago
DanielSank
16.8k44978
asked 5 hours ago
Wasserwaage
1175
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (the "external world") that acts as a generator of random phase. The introduction, however, is not quite clear to me:
The wave function $Psi$ of a truly isolated system may be expressed
as a linear superposition of a complete orthonormal set of stationary
wave functions $phi_n$
$$Psi=sum_n c_n phi_n$$
Here we can regard the system plus the external world as a truly
isolated system. The wave function $Psi$ for this whole system will
depend on both the coordinates of the system under consideration and
the coordinates of the external world. If $phi_n$ denotes a
complete set of orthonormal wave functions of the system, then $Psi$
is still formally given by the above equation, but the $c_n$ are to
be interpreted as wave functions of the external world.
Why is that? The definition of $c_n$is
$$c_n= langle phi_n|Psi rangle$$
which is a complex number. How do I get from there to the $c_n$ being wave functions of the outside world?
quantum-mechanics statistical-mechanics waves phase-space
edited 2 hours ago
DanielSank
16.8k44978
asked 5 hours ago
Wasserwaage
1175
In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (the "external world") that acts as a generator of random phase. The introduction, however, is not quite clear to me:
The wave function $Psi$ of a truly isolated system may be expressed
as a linear superposition of a complete orthonormal set of stationary
wave functions $phi_n$
$$Psi=sum_n c_n phi_n$$
Here we can regard the system plus the external world as a truly
isolated system. The wave function $Psi$ for this whole system will
depend on both the coordinates of the system under consideration and
the coordinates of the external world. If $phi_n$ denotes a
complete set of orthonormal wave functions of the system, then $Psi$
is still formally given by the above equation, but the $c_n$ are to
be interpreted as wave functions of the external world.
Why is that? The definition of $c_n$is
$$c_n= langle phi_n|Psi rangle$$
which is a complex number. How do I get from there to the $c_n$ being wave functions of the outside world?
quantum-mechanics statistical-mechanics waves phase-space
quantum-mechanics statistical-mechanics waves phase-space
edited 2 hours ago
DanielSank
16.8k44978
asked 5 hours ago
Wasserwaage
1175
edited 2 hours ago
DanielSank
16.8k44978
asked 5 hours ago
Wasserwaage
1175
edited 2 hours ago
DanielSank
16.8k44978
edited 2 hours ago
DanielSank
16.8k44978
edited 2 hours ago
DanielSank
16.8k44978
16.8k44978
asked 5 hours ago
Wasserwaage
1175
asked 5 hours ago
Wasserwaage
1175
asked 5 hours ago
Wasserwaage
1175
1175
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
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up vote
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Huang seems to conceal the fact that he splits the Hilbert space in two $mathcalH=mathcalH_SotimesmathcalH_E$, where $S$ is the system and $E$ is the envoirement. A general state in the total Hilber space can be written as
$$rvertPsirangle=sum_n,m gamma_nm rvertphi_nrangleotimesrvertpsi_mrangle$$ where $phi$ are a basis of $mathcalH_S$ and $psi$ are a basis of $mathcalH_E$ and where $gamma_nm$ are complex numbers defined as
$$gamma_nm=langle phi_notimespsi_mrvertPsirangle$$ in general, $Psi$ is an entangled state.
If you define $rvert c_nrangle= sum_m gamma_nm rvertpsi_mrangle$ then you can write
$$rvertPsirangle=sum_n rvert c_nrangleotimesrvertphi_nrangle$$
where the "coefficients" $c_n$ now are linear combinations of states describing the envoirement.
If you project onto the coordinate basis to obtain the wave functions you recover the Huang expression
$$Psi=sum_nc_n phi_n$$
where now $c_n=langle x_Ervert c_nrangle$ and $phi_n=langle x_Srvert phi_nrangle$
answered 1 hour ago
Fra
1,1451517
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Huang seems to conceal the fact that he splits the Hilbert space in two $mathcalH=mathcalH_SotimesmathcalH_E$, where $S$ is the system and $E$ is the envoirement. A general state in the total Hilber space can be written as
$$rvertPsirangle=sum_n,m gamma_nm rvertphi_nrangleotimesrvertpsi_mrangle$$ where $phi$ are a basis of $mathcalH_S$ and $psi$ are a basis of $mathcalH_E$ and where $gamma_nm$ are complex numbers defined as
$$gamma_nm=langle phi_notimespsi_mrvertPsirangle$$ in general, $Psi$ is an entangled state.
If you define $rvert c_nrangle= sum_m gamma_nm rvertpsi_mrangle$ then you can write
$$rvertPsirangle=sum_n rvert c_nrangleotimesrvertphi_nrangle$$
where the "coefficients" $c_n$ now are linear combinations of states describing the envoirement.
If you project onto the coordinate basis to obtain the wave functions you recover the Huang expression
$$Psi=sum_nc_n phi_n$$
where now $c_n=langle x_Ervert c_nrangle$ and $phi_n=langle x_Srvert phi_nrangle$
answered 1 hour ago
Fra
1,1451517
add a comment |Â
up vote
3
down vote
accepted
Huang seems to conceal the fact that he splits the Hilbert space in two $mathcalH=mathcalH_SotimesmathcalH_E$, where $S$ is the system and $E$ is the envoirement. A general state in the total Hilber space can be written as
$$rvertPsirangle=sum_n,m gamma_nm rvertphi_nrangleotimesrvertpsi_mrangle$$ where $phi$ are a basis of $mathcalH_S$ and $psi$ are a basis of $mathcalH_E$ and where $gamma_nm$ are complex numbers defined as
$$gamma_nm=langle phi_notimespsi_mrvertPsirangle$$ in general, $Psi$ is an entangled state.
If you define $rvert c_nrangle= sum_m gamma_nm rvertpsi_mrangle$ then you can write
$$rvertPsirangle=sum_n rvert c_nrangleotimesrvertphi_nrangle$$
where the "coefficients" $c_n$ now are linear combinations of states describing the envoirement.
If you project onto the coordinate basis to obtain the wave functions you recover the Huang expression
$$Psi=sum_nc_n phi_n$$
where now $c_n=langle x_Ervert c_nrangle$ and $phi_n=langle x_Srvert phi_nrangle$
answered 1 hour ago
Fra
1,1451517
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Huang seems to conceal the fact that he splits the Hilbert space in two $mathcalH=mathcalH_SotimesmathcalH_E$, where $S$ is the system and $E$ is the envoirement. A general state in the total Hilber space can be written as
$$rvertPsirangle=sum_n,m gamma_nm rvertphi_nrangleotimesrvertpsi_mrangle$$ where $phi$ are a basis of $mathcalH_S$ and $psi$ are a basis of $mathcalH_E$ and where $gamma_nm$ are complex numbers defined as
$$gamma_nm=langle phi_notimespsi_mrvertPsirangle$$ in general, $Psi$ is an entangled state.
If you define $rvert c_nrangle= sum_m gamma_nm rvertpsi_mrangle$ then you can write
$$rvertPsirangle=sum_n rvert c_nrangleotimesrvertphi_nrangle$$
where the "coefficients" $c_n$ now are linear combinations of states describing the envoirement.
If you project onto the coordinate basis to obtain the wave functions you recover the Huang expression
$$Psi=sum_nc_n phi_n$$
where now $c_n=langle x_Ervert c_nrangle$ and $phi_n=langle x_Srvert phi_nrangle$
answered 1 hour ago
Fra
1,1451517
Huang seems to conceal the fact that he splits the Hilbert space in two $mathcalH=mathcalH_SotimesmathcalH_E$, where $S$ is the system and $E$ is the envoirement. A general state in the total Hilber space can be written as
$$rvertPsirangle=sum_n,m gamma_nm rvertphi_nrangleotimesrvertpsi_mrangle$$ where $phi$ are a basis of $mathcalH_S$ and $psi$ are a basis of $mathcalH_E$ and where $gamma_nm$ are complex numbers defined as
$$gamma_nm=langle phi_notimespsi_mrvertPsirangle$$ in general, $Psi$ is an entangled state.
If you define $rvert c_nrangle= sum_m gamma_nm rvertpsi_mrangle$ then you can write
$$rvertPsirangle=sum_n rvert c_nrangleotimesrvertphi_nrangle$$
where the "coefficients" $c_n$ now are linear combinations of states describing the envoirement.
If you project onto the coordinate basis to obtain the wave functions you recover the Huang expression
$$Psi=sum_nc_n phi_n$$
where now $c_n=langle x_Ervert c_nrangle$ and $phi_n=langle x_Srvert phi_nrangle$
answered 1 hour ago
Fra
1,1451517
edited 1 hour ago
answered 1 hour ago
Fra
1,1451517
answered 1 hour ago
Fra
1,1451517
answered 1 hour ago
Fra
1,1451517
1,1451517
add a comment |Â
add a comment |Â
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