Probability density and kinetic

In summary: If we assume the speed in the first half is ##v## and in the second half ##c_1 v## with ##c_1 > 1## then a bounce from the "midpoint to back to the midpoint" takes ##T_1 =2L/v## and ##T_2 = 2L/(c_1v)## respectively.The probability of finding the particle in each place is then##P_1 =\frac{T_1}{T_1+T_2}= \frac{1}{1+1/c_1}##and##P_ 2= \frac{T_2}{T_1+T_2} =
  • #1
Incand
334
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energy.
Consider a particle in a box of the form
##V(x)=
\begin{cases}0 \; \; \; -L < x < 0\\
V_0 \; \; \; 0<x<L\\
\infty \; \; \; \text{ elsewhere}.\end{cases}##
One can show that the probability density
##P(x) = \Psi^* \Psi## is greater in the region of lower kinetic energy (that is at higher potential energy).

What's the physical explanation for this? My notes say something along the lines of "lower velocity give higher probability density" which seems very vague.
I'm guessing this is also related to how the Boltzmann distribution, where the probability decreases with the energy.
 
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  • #2
Is this a homework problem? Then post it in the homework forum. Please post a complete problem description. Otherwise we have to guess, what the problem is!

Otherwise, I guess they ask to find the energy eigenvalues and eigenfunctions and calculate the position-probability density distribution for these eigenstates. That's done by solving the time-independent Schrödinger equation.
 
  • #3
Incand said:
What's the physical explanation for this? My notes say something along the lines of "lower velocity give higher probability density" which seems very vague.
A lower kinetic energy means a lower speed, and a slower-moving particle takes longer to leave a given region... So intuitively you would expect the particle to spend more time in the regions of low kinetic energy just because once there it won't be in any hurry to leave.

It's worth taking a moment to calculate the classical trajectory of a classical particle bouncing back and forth between the walls of the box. In one cycle, how much time does the particle spend in the region between -L and 0, and how much time does the particle spend in the region between 0 and L? The ratio between the two determines the probability of finding the particle In either region at any randomly selected moment.
 
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  • #4
vanhees71 said:
Is this a homework problem? Then post it in the homework forum. Please post a complete problem description. Otherwise we have to guess, what the problem is!

Otherwise, I guess they ask to find the energy eigenvalues and eigenfunctions and calculate the position-probability density distribution for these eigenstates. That's done by solving the time-independent Schrödinger equation.
It's not a homework problem, I just gave an example to describe what I mean. I'm able to show the above result (that would have been a homework problem!), what I was wondering was the physical motivation behind it.

Nugatory said:
A lower kinetic energy means a lower speed, and a slower-moving particle takes longer to leave a given region... So intuitively you would expect the particle to spend more time in the regions of low kinetic just because once there it won't be in any hurry to leave.

It's worth taking a moment to calculate the classical trajectory of a classical particle bouncing back and forth between the walls of the box. In one cycle, how much time does the particle spend in the region between -L and 0, and how much time does the particle spend in the region between 0 and L? The ratio between the two determines the probability of finding the particle In either region at any randomly selected moment.

That makes sense, thanks! And the example nicely illustrates this as well:
If we assume the speed in the first half is ##v## and in the second half ##c_1 v## with ##c_1 > 1## then a bounce from the "midpoint to back to the midpoint" takes ##T_1 =2L/v## and ##T_2 = 2L/(c_1v)## respectively.
The probability of finding the particle in each place is then
##P_1 =\frac{T_1}{T_1+T_2}= \frac{1}{1+1/c_1}##and
##P_ 2= \frac{T_2}{T_1+T_2} = \frac{1}{1+c_1}##
so for ##c_1 >1## we have ##P_2 < P_1## so the particle clearly spends more time in region ##1##.

Edit: fixed a mistake, think it's correct now
 
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What is probability density?

Probability density is a measure of the likelihood of a continuous random variable taking on a specific value. It is represented by a curve on a graph, with the area under the curve representing the probability of the variable falling within a certain range of values.

How is probability density related to kinetic energy?

In physics, probability density is used to describe the distribution of kinetic energy among particles in a system. The more spread out the probability density is, the more evenly distributed the kinetic energy is among the particles.

What is the difference between probability density and probability?

Probability density is a continuous concept, while probability is a discrete concept. Probability density is used to describe the likelihood of a continuous random variable taking on a specific value, while probability is used to describe the likelihood of a discrete event occurring.

How is probability density calculated?

The calculation of probability density depends on the specific distribution of the random variable. For a continuous random variable, it is typically calculated using calculus by finding the area under the probability density curve.

What is the unit of measurement for probability density?

The unit of measurement for probability density depends on the specific units of the random variable. For example, if the random variable is measured in meters, the unit of measurement for probability density would be meters^-1. However, it is important to note that probability density is not a physical quantity and does not have a physical unit like energy or mass.

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