Non-Relativistic Limit of Klein-Gordon Probability Density
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
begin{eqnarray}
P & = & dfrac{ihbar}{2mc^2}left(Phi^*dfrac{partialPhi}{partial t}-Phidfrac{partialPhi^*}{partial t}right) \
vec{j} &=& dfrac{hbar}{2mi}left(Phi^*vec{nabla}Phi-Phivec{nabla}Phi^*right)
end{eqnarray}
$$
together with the statement that:
One can show that in the non-relativistic limit, the known expressions for the probability density and current are recovered.
The 'known' expressions are:
$$
begin{eqnarray}
rho &=& Psi^*Psi \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
When taking a 'non-relativistic limit', I am used to taking the limit $c to infty$, which does give the right result for $vec{j}$, but for the density produces $P=0$. How should one then take said limit to recover the non-relativistic equations?
quantum-mechanics wavefunction speed-of-light schroedinger-equation klein-gordon-equation
add a comment |
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
begin{eqnarray}
P & = & dfrac{ihbar}{2mc^2}left(Phi^*dfrac{partialPhi}{partial t}-Phidfrac{partialPhi^*}{partial t}right) \
vec{j} &=& dfrac{hbar}{2mi}left(Phi^*vec{nabla}Phi-Phivec{nabla}Phi^*right)
end{eqnarray}
$$
together with the statement that:
One can show that in the non-relativistic limit, the known expressions for the probability density and current are recovered.
The 'known' expressions are:
$$
begin{eqnarray}
rho &=& Psi^*Psi \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
When taking a 'non-relativistic limit', I am used to taking the limit $c to infty$, which does give the right result for $vec{j}$, but for the density produces $P=0$. How should one then take said limit to recover the non-relativistic equations?
quantum-mechanics wavefunction speed-of-light schroedinger-equation klein-gordon-equation
1
For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein.
– Qmechanic♦
5 hours ago
add a comment |
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
begin{eqnarray}
P & = & dfrac{ihbar}{2mc^2}left(Phi^*dfrac{partialPhi}{partial t}-Phidfrac{partialPhi^*}{partial t}right) \
vec{j} &=& dfrac{hbar}{2mi}left(Phi^*vec{nabla}Phi-Phivec{nabla}Phi^*right)
end{eqnarray}
$$
together with the statement that:
One can show that in the non-relativistic limit, the known expressions for the probability density and current are recovered.
The 'known' expressions are:
$$
begin{eqnarray}
rho &=& Psi^*Psi \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
When taking a 'non-relativistic limit', I am used to taking the limit $c to infty$, which does give the right result for $vec{j}$, but for the density produces $P=0$. How should one then take said limit to recover the non-relativistic equations?
quantum-mechanics wavefunction speed-of-light schroedinger-equation klein-gordon-equation
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
begin{eqnarray}
P & = & dfrac{ihbar}{2mc^2}left(Phi^*dfrac{partialPhi}{partial t}-Phidfrac{partialPhi^*}{partial t}right) \
vec{j} &=& dfrac{hbar}{2mi}left(Phi^*vec{nabla}Phi-Phivec{nabla}Phi^*right)
end{eqnarray}
$$
together with the statement that:
One can show that in the non-relativistic limit, the known expressions for the probability density and current are recovered.
The 'known' expressions are:
$$
begin{eqnarray}
rho &=& Psi^*Psi \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
When taking a 'non-relativistic limit', I am used to taking the limit $c to infty$, which does give the right result for $vec{j}$, but for the density produces $P=0$. How should one then take said limit to recover the non-relativistic equations?
quantum-mechanics wavefunction speed-of-light schroedinger-equation klein-gordon-equation
quantum-mechanics wavefunction speed-of-light schroedinger-equation klein-gordon-equation
edited 5 hours ago
Qmechanic♦
101k121821140
101k121821140
asked 5 hours ago
Simon
755313
755313
1
For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein.
– Qmechanic♦
5 hours ago
add a comment |
1
For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein.
– Qmechanic♦
5 hours ago
1
1
For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein.
– Qmechanic♦
5 hours ago
For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein.
– Qmechanic♦
5 hours ago
add a comment |
2 Answers
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You can substitute $Phi = e^{-mc^2t/hbar} Psi$ and then neglect the second order time derivative of $Psi$. Drop the constant $mc^2$ and you will have recovered the Schrödinger equation.
add a comment |
The trick is to make the approach for the relativistic Klein-Gordon wave function
$$ Phi(vec{r},t) = Psi(vec{r},t) e^{-frac{imc^2t}{hbar}} $$
(The physical reasoning behind this approach is:
The fast oscillating exponential is the solution for the particle at rest.
And compared to that, $Psi$ will give only slow variations.)
From that you find its derivatives
$$
begin{eqnarray}
frac{partialPhi}{partial t} &=& left( frac{partialPsi}{partial t} - frac{imc^2}{hbar}Psi right) e^{-frac{imc^2t}{hbar}} \
vec{nabla}Phi &=& vec{nabla}Psi e^{-frac{imc^2t}{hbar}}
end{eqnarray}
$$
Plug these into the definitions of the Klein-Gordon probability density and current ($P$ and $vec{j}$) and you get
$$
begin{eqnarray}
P &=& Psi^* Psi + frac{ihbar}{2mc^2} left( Psi^*frac{partialPsi}{partial t} - Psi frac{partialPsi^*}{partial t} right) \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
Now you can do the limit $c to infty$ and get the 'known' expressions of the non-relativistic Schrödinger probability density and current.
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
You can substitute $Phi = e^{-mc^2t/hbar} Psi$ and then neglect the second order time derivative of $Psi$. Drop the constant $mc^2$ and you will have recovered the Schrödinger equation.
add a comment |
You can substitute $Phi = e^{-mc^2t/hbar} Psi$ and then neglect the second order time derivative of $Psi$. Drop the constant $mc^2$ and you will have recovered the Schrödinger equation.
add a comment |
You can substitute $Phi = e^{-mc^2t/hbar} Psi$ and then neglect the second order time derivative of $Psi$. Drop the constant $mc^2$ and you will have recovered the Schrödinger equation.
You can substitute $Phi = e^{-mc^2t/hbar} Psi$ and then neglect the second order time derivative of $Psi$. Drop the constant $mc^2$ and you will have recovered the Schrödinger equation.
answered 2 hours ago
my2cts
4,4482617
4,4482617
add a comment |
add a comment |
The trick is to make the approach for the relativistic Klein-Gordon wave function
$$ Phi(vec{r},t) = Psi(vec{r},t) e^{-frac{imc^2t}{hbar}} $$
(The physical reasoning behind this approach is:
The fast oscillating exponential is the solution for the particle at rest.
And compared to that, $Psi$ will give only slow variations.)
From that you find its derivatives
$$
begin{eqnarray}
frac{partialPhi}{partial t} &=& left( frac{partialPsi}{partial t} - frac{imc^2}{hbar}Psi right) e^{-frac{imc^2t}{hbar}} \
vec{nabla}Phi &=& vec{nabla}Psi e^{-frac{imc^2t}{hbar}}
end{eqnarray}
$$
Plug these into the definitions of the Klein-Gordon probability density and current ($P$ and $vec{j}$) and you get
$$
begin{eqnarray}
P &=& Psi^* Psi + frac{ihbar}{2mc^2} left( Psi^*frac{partialPsi}{partial t} - Psi frac{partialPsi^*}{partial t} right) \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
Now you can do the limit $c to infty$ and get the 'known' expressions of the non-relativistic Schrödinger probability density and current.
add a comment |
The trick is to make the approach for the relativistic Klein-Gordon wave function
$$ Phi(vec{r},t) = Psi(vec{r},t) e^{-frac{imc^2t}{hbar}} $$
(The physical reasoning behind this approach is:
The fast oscillating exponential is the solution for the particle at rest.
And compared to that, $Psi$ will give only slow variations.)
From that you find its derivatives
$$
begin{eqnarray}
frac{partialPhi}{partial t} &=& left( frac{partialPsi}{partial t} - frac{imc^2}{hbar}Psi right) e^{-frac{imc^2t}{hbar}} \
vec{nabla}Phi &=& vec{nabla}Psi e^{-frac{imc^2t}{hbar}}
end{eqnarray}
$$
Plug these into the definitions of the Klein-Gordon probability density and current ($P$ and $vec{j}$) and you get
$$
begin{eqnarray}
P &=& Psi^* Psi + frac{ihbar}{2mc^2} left( Psi^*frac{partialPsi}{partial t} - Psi frac{partialPsi^*}{partial t} right) \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
Now you can do the limit $c to infty$ and get the 'known' expressions of the non-relativistic Schrödinger probability density and current.
add a comment |
The trick is to make the approach for the relativistic Klein-Gordon wave function
$$ Phi(vec{r},t) = Psi(vec{r},t) e^{-frac{imc^2t}{hbar}} $$
(The physical reasoning behind this approach is:
The fast oscillating exponential is the solution for the particle at rest.
And compared to that, $Psi$ will give only slow variations.)
From that you find its derivatives
$$
begin{eqnarray}
frac{partialPhi}{partial t} &=& left( frac{partialPsi}{partial t} - frac{imc^2}{hbar}Psi right) e^{-frac{imc^2t}{hbar}} \
vec{nabla}Phi &=& vec{nabla}Psi e^{-frac{imc^2t}{hbar}}
end{eqnarray}
$$
Plug these into the definitions of the Klein-Gordon probability density and current ($P$ and $vec{j}$) and you get
$$
begin{eqnarray}
P &=& Psi^* Psi + frac{ihbar}{2mc^2} left( Psi^*frac{partialPsi}{partial t} - Psi frac{partialPsi^*}{partial t} right) \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
Now you can do the limit $c to infty$ and get the 'known' expressions of the non-relativistic Schrödinger probability density and current.
The trick is to make the approach for the relativistic Klein-Gordon wave function
$$ Phi(vec{r},t) = Psi(vec{r},t) e^{-frac{imc^2t}{hbar}} $$
(The physical reasoning behind this approach is:
The fast oscillating exponential is the solution for the particle at rest.
And compared to that, $Psi$ will give only slow variations.)
From that you find its derivatives
$$
begin{eqnarray}
frac{partialPhi}{partial t} &=& left( frac{partialPsi}{partial t} - frac{imc^2}{hbar}Psi right) e^{-frac{imc^2t}{hbar}} \
vec{nabla}Phi &=& vec{nabla}Psi e^{-frac{imc^2t}{hbar}}
end{eqnarray}
$$
Plug these into the definitions of the Klein-Gordon probability density and current ($P$ and $vec{j}$) and you get
$$
begin{eqnarray}
P &=& Psi^* Psi + frac{ihbar}{2mc^2} left( Psi^*frac{partialPsi}{partial t} - Psi frac{partialPsi^*}{partial t} right) \
vec{j} &=& dfrac{hbar}{2mi}left(Psi^*vec{nabla}Psi-Psivec{nabla}Psi^*right)
end{eqnarray}
$$
Now you can do the limit $c to infty$ and get the 'known' expressions of the non-relativistic Schrödinger probability density and current.
edited 11 mins ago
answered 1 hour ago
Thomas Fritsch
18419
18419
add a comment |
add a comment |
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1
For a connection between Schr. eq. and Klein-Gordon eq, see e.g. A. Zee, QFT in a Nutshell, Chap. III.5, and this Phys.SE post plus links therein.
– Qmechanic♦
5 hours ago