An electron has a wavefunction ψ(x)=C e^-|x| / x0 where x0 is a constant and C=1 / √(x0) for nor

An electron has a wavefunction ψ(x)=C e^-|x| / x0 where x0...

An electron has a wavefunction $$ \psi(x)=C e^{-|x| / x_{0}} $$ where $x_{0}$ is a constant and $C=1 / \sqrt{x_{0}}$ for normalization (see Example $6.1) .$ For this case, obtain expressions for $\langle x\rangle$ and $\Delta x$ in terms of $x_{0} .$ Also calculate the probability that the electron will be found within a standard deviation of its average position, that is, in the range $\langle x\rangle-\Delta x$ to $\langle x\rangle+\Delta x,$ and show that this is independent of $x_{0}.$

An electron has a wavefunction
$$
\psi(x)=C e^{-|x| / x_{0}}
$$
where $x_{0}$ is a constant and $C=1 / \sqrt{x_{0}}$ for normalization (see Example $6.1) .$ For this case, obtain expressions for $\langle x\rangle$ and $\Delta x$ in terms of $x_{0} .$ Also calculate the probability that the electron will be found within a standard deviation of its average position, that is, in the range $\langle x\rangle-\Delta x$ to $\langle x\rangle+\Delta x,$ and show that this is independent of $x_{0}.$

Key Concepts

Standard Deviation

The standard deviation in quantum mechanics quantifies the uncertainty or spread in the possible outcomes of a measurement. For position, it is calculated as the square root of the difference between the expectation value of the square of the position and the square of the expectation value of the position.

Wavefunction

In quantum mechanics, the wavefunction represents the probability amplitude for a particle's state. Its square modulus gives the probability density, indicating where the particle is likely to be found when a measurement is taken.

Normalization

Normalization is the requirement that the total probability of finding the particle somewhere in space is equal to one. This is achieved by choosing the appropriate constant such that the integral of the square modulus of the wavefunction over all space equals unity.

Expectation Value

The expectation value of an observable, like position, is the average value obtained from many measurements. It is computed by integrating the product of the observable and the probability density over all space.

Probability Calculation

Probability calculation in this context involves integrating the probability density over a specific interval—in this case, within one standard deviation of the mean position—to determine the likelihood that a measurement falls within that range.

Transcript

00:01 So this is a wave function that we're given for this question, describing an electron.

00:07 Now we want to sketch the wave function.

00:09 Now we just focus on first the positive x direction.

00:15 So in a positive x direction, when x is near to 0, we get this exponential term going to 1, and so the wave function actually goes to a.

00:31 So it starts from a and then as it moves as x increases we get an exponentially decaying curve as such so it's exponentially decaying same for the negative side so if you consider the negative x part again we start off when x is near to 0 this will be this exponential term is 1 so it starts off at a and then it negatively, when it moves to the negative x side, you know, we get a exponential decaying function, because it's negative a times that, because x is a negative value, so it will be negatively decaying.

01:27 So this is our wave function.

01:30 Now you want to find the sketch the probability density.

01:34 So the probability density is just side.

01:37 Square.

01:44 Now for this curve, what we do is we are just squaring the magnitude, right, so the curve, because what we will get is basically a square times e to power negative 2a x and a square equals times e to power of positive 2a x.

02:09 For x more than 0, and x less than zero respectively.

02:17 So the only thing that change is probably the highest magnitude over here is now a square.

02:26 But what we still get is an exponentially decaying curve on both sides of the axis, right, and it's symmetrical as well.

02:37 So this is for the probably that's the function.

02:45 Now of course the entire function is not fully differentiable because of this discontinuity at the x equals to 0 right over here 0 point there's this sharp turn which makes it non not differentiable but it is still a good way function or a physically reasonable way function because of two reasons right one is because it exponentially decays, decays to 0, when x goes to positive and negative infinity.

03:32 And what this means is it implies that we can actually normalize, you can normalize our wave function because it won't blow up to infinity when we do an integration throughout all space.

03:52 Another thing would be that this function is continuous, there's no discontinuity at the center, even though there is a sharp kink, it is still continuous.

04:16 There's no break in the wave function, so it's continuous and therefore it is a possible, a reasonable wave function.

04:25 So there's two different arguments.

04:30 Now to actually normalize this, we have to do the integration.

04:34 Remember that the definition for the normalization condition is that integrating about all space must give us a total probability of one...

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