I quantum particle has a wave function ψ(x)={ √((2)/(a)) e^-x / a for x > 0 0 for

I quantum particle has a wave function ψ(x)={ √((2)/(a))...

I quantum particle has a wave function $$ \psi(x)=\left\{\begin{array}{ll}{\sqrt{\frac{2}{a}} e^{-x / a}} & {\text { for } x > 0} \\ {0} & {\text { for } x < 0}\end{array}\right\} $$ (a) Find and sketch the probability density. (b) Find the probability that the particle will be at any point where $x<0 .$ (c) Show that $\psi$ is normalized and then find the probability of finding the particle between $x=0$ and $x=a$ .

I quantum particle has a wave function
$$
\psi(x)=\left\{\begin{array}{ll}{\sqrt{\frac{2}{a}} e^{-x / a}} & {\text { for } x > 0} \\ {0} & {\text { for } x < 0}\end{array}\right\}
$$
(a) Find and sketch the probability density. (b) Find the probability that the particle will be at any point where $x<0 .$ (c) Show that $\psi$ is normalized and then find the probability of finding the particle between $x=0$ and $x=a$ .

Key Concepts

Quantum Wave Function

The quantum wave function is a complex function that contains all the information about the quantum state of a particle. It is used to calculate probabilities and expectation values for various observables, by taking its absolute square to obtain the probability density.

Probability Density

The probability density is given by the absolute square of the wave function, |?(x)|², and represents the probability per unit length (in one dimension) of finding the particle at a particular position. It is a fundamental concept in quantum mechanics for interpreting the results of measurements.

Normalization

Normalization is the process of ensuring that the total probability of finding the particle somewhere in space is equal to one. This is achieved by integrating the probability density over all space and adjusting any constants in the wave function so that the integral equals one.

Integration in Probability Calculations

Integration is used to calculate the probability of finding the particle within a certain region of space by integrating the probability density function over that region. It is a key mathematical method in quantum mechanics when dealing with continuous probability distributions.

Piecewise Defined Functions

Piecewise defined functions in quantum mechanics occur when the wave function is defined differently in separate spatial regions. Understanding how to handle such functions is important, as it requires separate integrations or evaluations in each region while ensuring overall consistency and normalization of the wave function.

Transcript

00:01 So the wave function that we're given over here, actually it's kind of like it only exists in the positive x value, and it's 0 for the negative x values.

00:14 You want to find the probability density, and probability density is simply the absolute square of the wave function, which would give us x over a exponential negative 2x over a for x more than 0 and still 0 because 0 squared 0 for x less than 0 want to sketch this out so let's sketch for the negative x values it's very simple it's just 0 all the way but when it hits the 0 point when x equals to 0 now the value increases right at this particular point when sorry at this point when x is wax becomes positive right the when we substitute in x equals to 0 into this expression our exponential term goes to 1 and what we're left with is just 2 over 8 so the amplitude or the pick at x equals to 0 is to over a and from there on it actually exponentially decays right this is a exponential negative x term so it will exponentially decay towards zero but never reaching zero only reaching zero at infinity right so now we are asked to show or to find what is the probability of the particle at any point or finding the particle at any point for negative x values.

02:25 So once again we know that the probability density is 0 for all negative values.

02:33 So to find x at any negative point, we can integrate from negative infinity to 0 the probability density function.

02:51 And we will see that because this is zero, we're integrating any zero value, which is just zero.

02:59 There's no probability of finding a particle at any point when x is negative.

03:08 Now to show that our wave function is normalized, we use the normalization condition where we integrate the probability density function across all space from negative, to infinity, the total probability must give us 1.

03:28 Now we have established that when we integrate from negative infinity to 0, it's 0.

03:35 So we can simplify this integral to just from 0 to positive infinity...

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