Integration by parts in proof of special case of Ehrenfest's theorem
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Our physics professor proved a special case of Ehrenfest's theorem, or to be more precise:
$$mlangle xrangle = langle prangle,$$
where we have used the definition:
$$langle xrangle = int_{mathbb{R}}psi^* (x,t), x,psi(x,t), dx,$$
and $psi$ is a normalized wave function.
In the proof at some point he writes the following:
$$begin{align}begin{aligned} imath frac { hbar } { 2 m } int _ { mathbb { R } } d x, x, Psi ( x , t ) ^ { * } &frac { partial ^ { 2 } } { partial x ^ { 2 } } Psi ( x , t ) = imath frac { hbar } { 2 m } left[ x Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right] _ { x = - infty } ^ { x = infty } \ & - imath frac { hbar } { 2 m } int _ { mathbb { R } } d x left( Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) + x frac { partial } { partial x } Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right). end{aligned}end{align}$$
I really don't understand how the integration by parts works here and would be really happy if someone could explain this.
integration definite-integrals physics
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
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add a comment |
$begingroup$
Our physics professor proved a special case of Ehrenfest's theorem, or to be more precise:
$$mlangle xrangle = langle prangle,$$
where we have used the definition:
$$langle xrangle = int_{mathbb{R}}psi^* (x,t), x,psi(x,t), dx,$$
and $psi$ is a normalized wave function.
In the proof at some point he writes the following:
$$begin{align}begin{aligned} imath frac { hbar } { 2 m } int _ { mathbb { R } } d x, x, Psi ( x , t ) ^ { * } &frac { partial ^ { 2 } } { partial x ^ { 2 } } Psi ( x , t ) = imath frac { hbar } { 2 m } left[ x Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right] _ { x = - infty } ^ { x = infty } \ & - imath frac { hbar } { 2 m } int _ { mathbb { R } } d x left( Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) + x frac { partial } { partial x } Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right). end{aligned}end{align}$$
I really don't understand how the integration by parts works here and would be really happy if someone could explain this.
integration definite-integrals physics
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
$endgroup$
1
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You integrate $frac{partial^2 }{partial x^2}psi(x,t)$ and differentiate $xpsi (x,t)^*$ with respect to $x$.
$endgroup$
– Lorenzo Quarisa
Jan 17 at 20:44
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What do you not understand about integrating by parts here? Let $u=xPsi^{*}(x,t)$ and $v=Psi_x(x,t)$. Also use the fact that the wave function is of Compact Support.
$endgroup$
– Mark Viola
Jan 17 at 21:17
add a comment |
$begingroup$
Our physics professor proved a special case of Ehrenfest's theorem, or to be more precise:
$$mlangle xrangle = langle prangle,$$
where we have used the definition:
$$langle xrangle = int_{mathbb{R}}psi^* (x,t), x,psi(x,t), dx,$$
and $psi$ is a normalized wave function.
In the proof at some point he writes the following:
$$begin{align}begin{aligned} imath frac { hbar } { 2 m } int _ { mathbb { R } } d x, x, Psi ( x , t ) ^ { * } &frac { partial ^ { 2 } } { partial x ^ { 2 } } Psi ( x , t ) = imath frac { hbar } { 2 m } left[ x Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right] _ { x = - infty } ^ { x = infty } \ & - imath frac { hbar } { 2 m } int _ { mathbb { R } } d x left( Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) + x frac { partial } { partial x } Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right). end{aligned}end{align}$$
I really don't understand how the integration by parts works here and would be really happy if someone could explain this.
integration definite-integrals physics
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
$endgroup$
Our physics professor proved a special case of Ehrenfest's theorem, or to be more precise:
$$mlangle xrangle = langle prangle,$$
where we have used the definition:
$$langle xrangle = int_{mathbb{R}}psi^* (x,t), x,psi(x,t), dx,$$
and $psi$ is a normalized wave function.
In the proof at some point he writes the following:
$$begin{align}begin{aligned} imath frac { hbar } { 2 m } int _ { mathbb { R } } d x, x, Psi ( x , t ) ^ { * } &frac { partial ^ { 2 } } { partial x ^ { 2 } } Psi ( x , t ) = imath frac { hbar } { 2 m } left[ x Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right] _ { x = - infty } ^ { x = infty } \ & - imath frac { hbar } { 2 m } int _ { mathbb { R } } d x left( Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) + x frac { partial } { partial x } Psi ( x , t ) ^ { * } frac { partial } { partial x } Psi ( x , t ) right). end{aligned}end{align}$$
I really don't understand how the integration by parts works here and would be really happy if someone could explain this.
integration definite-integrals physics
integration definite-integrals physics
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
asked Jan 17 at 20:40
Marius JaegerMarius Jaeger
61
61
1
$begingroup$
You integrate $frac{partial^2 }{partial x^2}psi(x,t)$ and differentiate $xpsi (x,t)^*$ with respect to $x$.
$endgroup$
– Lorenzo Quarisa
Jan 17 at 20:44
$begingroup$
What do you not understand about integrating by parts here? Let $u=xPsi^{*}(x,t)$ and $v=Psi_x(x,t)$. Also use the fact that the wave function is of Compact Support.
$endgroup$
– Mark Viola
Jan 17 at 21:17
add a comment |
1
$begingroup$
You integrate $frac{partial^2 }{partial x^2}psi(x,t)$ and differentiate $xpsi (x,t)^*$ with respect to $x$.
$endgroup$
– Lorenzo Quarisa
Jan 17 at 20:44
$begingroup$
What do you not understand about integrating by parts here? Let $u=xPsi^{*}(x,t)$ and $v=Psi_x(x,t)$. Also use the fact that the wave function is of Compact Support.
$endgroup$
– Mark Viola
Jan 17 at 21:17
1
1
$begingroup$
You integrate $frac{partial^2 }{partial x^2}psi(x,t)$ and differentiate $xpsi (x,t)^*$ with respect to $x$.
$endgroup$
– Lorenzo Quarisa
Jan 17 at 20:44
$begingroup$
You integrate $frac{partial^2 }{partial x^2}psi(x,t)$ and differentiate $xpsi (x,t)^*$ with respect to $x$.
$endgroup$
– Lorenzo Quarisa
Jan 17 at 20:44
$begingroup$
What do you not understand about integrating by parts here? Let $u=xPsi^{*}(x,t)$ and $v=Psi_x(x,t)$. Also use the fact that the wave function is of Compact Support.
$endgroup$
– Mark Viola
Jan 17 at 21:17
$begingroup$
What do you not understand about integrating by parts here? Let $u=xPsi^{*}(x,t)$ and $v=Psi_x(x,t)$. Also use the fact that the wave function is of Compact Support.
$endgroup$
– Mark Viola
Jan 17 at 21:17
add a comment |
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$begingroup$
You integrate $frac{partial^2 }{partial x^2}psi(x,t)$ and differentiate $xpsi (x,t)^*$ with respect to $x$.
$endgroup$
– Lorenzo Quarisa
Jan 17 at 20:44
$begingroup$
What do you not understand about integrating by parts here? Let $u=xPsi^{*}(x,t)$ and $v=Psi_x(x,t)$. Also use the fact that the wave function is of Compact Support.
$endgroup$
– Mark Viola
Jan 17 at 21:17