Let Pa b(t) be the probability of finding a particle in the...

Let $P_{a b}(t)$ be the probability of finding a particle in the range $(a<x<b)$, at time $t$ (a) Show that $$ \frac{d P_{a b}}{d t}=J(a, t)-J(b, t) $$ where $$ J(x, t) \equiv \frac{i \hbar}{2 m}\left(\Psi \frac{\partial \Psi^{*}}{\partial x}-\Psi^{*} \frac{\partial \Psi}{\partial x}\right) $$ What are the units of $J(x, t) ?$ Cornment: $J$ is called the probability current, because it tells you the rate at which probability is "flowing" past the point $x$. If $P_{a b}(t)$ is increasing, then more probability is flowing into the region at one end than flows out at the other. (b) Find the probability current for the wave function in Problem $1.9 .$ (This is not a very pithy example, I'm afraid; we'll encounter more substantial ones in due course.)

Let $P_{a b}(t)$ be the probability of finding a particle in the range $(a<x<b)$, at time $t$
(a) Show that
$$
\frac{d P_{a b}}{d t}=J(a, t)-J(b, t)
$$
where
$$
J(x, t) \equiv \frac{i \hbar}{2 m}\left(\Psi \frac{\partial \Psi^{*}}{\partial x}-\Psi^{*} \frac{\partial \Psi}{\partial x}\right)
$$
What are the units of $J(x, t) ?$ Cornment: $J$ is called the probability current, because it tells you the rate at which probability is "flowing" past the point $x$. If $P_{a b}(t)$ is increasing, then more probability is flowing into the region at one end than flows out at the other.
(b) Find the probability current for the wave function in Problem $1.9 .$ (This is not a very pithy example, I'm afraid; we'll encounter more substantial ones in due course.)

Key Concepts

Dimensional Analysis

Dimensional analysis involves checking the consistency of units in physical expressions. In the context of the probability current, it helps confirm that the derived expression for the current has the correct units. Since probability is dimensionless and its rate of change has units of inverse time, the probability current must also carry units of inverse time. This analysis is essential for verifying the physical plausibility of equations in quantum mechanics.

Wave Function

The wave function is a fundamental quantity in quantum mechanics that encodes the state of a system. Its absolute square gives the probability density of finding a particle at a given position and time. The wave function, along with its complex conjugate and derivatives, appears in the formulation of the probability current and underpins many quantum phenomena.

Probability Current

The probability current is a concept in quantum mechanics that quantifies the rate at which probability density flows through a given point in space. It is defined in terms of the wave function and its spatial derivative, and it plays a central role in describing how probability is conserved over time, analogous to current density in fluid dynamics or electromagnetism.

Continuity Equation

The continuity equation in quantum mechanics expresses the local conservation of probability. It relates the time derivative of the probability density in a region to the divergence (or difference in one dimension) of the probability current. This equation ensures that the total probability is conserved over time, linking the change in the probability inside a region to the net flow of probability into or out of that region.

Transcript

00:01 This is the question number 14 and in this question the probability of finding a particle in the range a and b at time t is pab that is the probability is time dependent and we have to show that the derivative of this probability is equal to the j -at minus j b t where j is the current range here let's define this probability first.

00:37 This is problem number 14.

00:47 And here this is some range and suppose this is x and these are two points that is a and b and the pab that is probability in this range is time dependent and we have to find the derivative of this property and the jxt is given which is ih bar by 2m and si del si complex conjugate del x minus si complice conjugate and then x minus si complex conjugate and then say by del x and we have to show that d p.

01:49 B by d t is equal to j a t minus j b t and we have to show this so let us define the probability in this range and which is defined as pab and which is time dependent is equal to a to b and the d x and mod of psi x t full square and if you we find the derivative of this that is d by d t of this b that is t is equal to a to b and del by del t s s x t square and d x here notice that this is the or time derivative and the integration is with respect to dx let's represent this equation by one and we have that is derivative of this that is del upon del t and this part can be as si star x t and si and it will be that is deli that is del s x by del t and si plus si star and del si by del si by del t so this is equation number two we have to find out this first and then substitute this value in this equation number one we have the sereniger equation that is i h bar del sci by del t is equal to minus h squared by 2m and del 2xxxx2x plus v.

04:54 And here this is the function of x and t and this is also function of x and t.

05:03 We have to find on this value del x x by del t and then take the complex conjugate.

05:10 So here this value will be del si by del t.

05:14 Del t is equal to and this will be here h bar as bar cancel and this will be i h bar by 2m and del 2 psi by del x square and this value upon v si upon i as bar if i multiply by i then it will be minus i by h bar and v si si x t if i take the complex conjugate of this then we will get del si by del t is equal to minus i h bar by two m and del two si complex conjugate by del x square plus i by h bar and here is the real that v si x t now find out this value that is del upon del t or del by del t mod of si x t square is equal to let's write this again so here this is del si by del t and si by d t and si the multiple of this that means this equation will be minus i h bar del to si by del h square and si plus i by h bar v si and si and now second term plus si star and del si by del t plus si star and del si by del t plus si and here it is it is plus i .h bar here one term is machine that is i .m.

07:56 And this is i h bar by 2m del 2xx2x2 by del x and this is minus i by h bar and v and si.

08:15 And si so here you can notice and this term is cancelled and then here you will get minus ih bar or we can take this positive here that is ih bar by 2m and here this si star del to si by del x square and this is minus del to si by del x square and si star and here we can write this that is i as bar by 2m and it can be written a del by del x and si star and del x by del x by del x by del x by del x and this is given in the column that with j x x by del x is defined that is i s bar by 2m, si d 'is star and x x star and by del x minus i this...

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