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Lendion
Lendion
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i would like to evaluate $$\int\mathcal{D}x\ e^{\frac{i}{\hbar}\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$$$\int\mathcal{D}x\ e^{-\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euler Lagrange equations for $$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

i would like to evaluate $$\int\mathcal{D}x\ e^{\frac{i}{\hbar}\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euler Lagrange equations for $$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

i would like to evaluate $$\int\mathcal{D}x\ e^{-\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euler Lagrange equations for $$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

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Qmechanic ♦
Qmechanic ♦
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how How to use saddle point approximation with path integrals?

i would like to evaluate $\int\mathcal{D}x\ e^{\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}$$$\int\mathcal{D}x\ e^{\frac{i}{\hbar}\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the EuilerEuler Lagrange equations for $\mathcal{L}=(\dot x+ \alpha x)^2$$$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

how to use saddle point approximation with path integrals

i would like to evaluate $\int\mathcal{D}x\ e^{\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euiler Lagrange equations for $\mathcal{L}=(\dot x+ \alpha x)^2$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

How to use saddle point approximation with path integrals?

i would like to evaluate $$\int\mathcal{D}x\ e^{\frac{i}{\hbar}\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}\tag{1}$$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euler Lagrange equations for $$\mathcal{L}=(\dot x+ \alpha x)^2\tag{2}$$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?

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Lendion
Lendion
  • 31
  • 1
  • 1
  • 4

how to use saddle point approximation with path integrals

i would like to evaluate $\int\mathcal{D}x\ e^{\int\limits_{-\infty}^{\infty} dt\ (\dot x+\alpha x)^2}$ and it is my understanding that the way to do so is using the saddle point approximation, however I'm new to using it and am unsure how to do so for the case of an action

i was able to solve the Euiler Lagrange equations for $\mathcal{L}=(\dot x+ \alpha x)^2$ however i am not sure of the next step

usually i would Taylor expand the function inside the exponent and evaluate the resulting gaussian integral but i am not sure if this requires me to take the second derivative of my action with respect to $x$ (or $\dot x$ as well?)

how do i proceed?