Expectation Value for a Function with Cusp

• Septim
In summary: Psi(x) +2a^{-3}x\Psi'(x).In summary, the conversation is about a homework assignment where the student is asked to find the expectation value of momentum squared in configuration space. They are struggling with expressing the first and second derivatives of the given wavefunction in terms of the signum function and Dirac delta function. They have also attached the original worksheet for reference. Another person in the conversation points out that the second derivative should be continuous and suggests doing the calculation in two parts for positive and negative x values. The student then clarifies that the first derivative has a jump at x=0 and the second derivative is infinite at x=0, and they
Septim

Homework Statement

In my homework assignment I have a wavefunction defined as $\Psi(x)=N\exp(-|x|/a)$ and I am asked to find the expectation value of momentum squared in configuration space.

Homework Equations

$\int\Psi*(x)\hat{p^2}\Psi(x)dx$

The Attempt at a Solution

N is $1/\sqrt{a}$ due to the normalization requirement.The first derivative of the wavefunction is piecewise defined hence the second derivative is discontinuous at x=0. I am having difficulties in expressing the first derivative and second derivative in terms of signum function and Dirac delta function. I would like to express them as analytically as possible but due to my unfamiliarity with these functions I am unable to do so. If I disregard the discontinuity of the first derivative function at x=0 the expectation value turns out to be negative. How can I overcome this situation?

Thanks for the replies.
Edit: I have attached the original worksheet in case you would like to look at it.

Attachments

• 20112-300-hw-2.pdf
23.8 KB · Views: 174
Hang on, isn't the second derivative continuous? Differentiating twice brings down two signs of x, so they cancel and limit is same from up and down. If the calculation gives you trouble, just do it in two parts, first for positive and then negative x.

The first derivative has a jump at x=0 and the second derivative is infinite at x=0. My question is that how to deal with this situation analytically by using Dirac Delta function and Unit step function.

Can you reproduce your differentiation steps here? You are making some mistake, because the second derivative of psi is clearly $a^{-2} \Psi$

1. What is the definition of expectation value for a function with cusp?

The expectation value for a function with cusp is a mathematical concept used in quantum mechanics to describe the average value of a physical quantity for a given wave function. It is calculated by taking the integral of the function multiplied by the probability density function, and can provide insight into the behavior of a system.

2. How is expectation value for a function with cusp different from a regular expectation value?

The difference between expectation value for a function with cusp and a regular expectation value is that the former takes into account the discontinuity or "cusp" in the function, while the latter assumes a continuous function. This can result in different values for the two types of expectation values.

3. Why is the expectation value for a function with cusp important in quantum mechanics?

In quantum mechanics, the expectation value for a function with cusp is important because it allows us to make predictions about the behavior of a system. By calculating the average value of a physical quantity, we can gain insight into the state of a system and make predictions about its future behavior.

4. How is the expectation value for a function with cusp used in practical applications?

The expectation value for a function with cusp is used in a variety of practical applications in quantum mechanics. It can be used to calculate the average energy of a system, the average position of a particle, and other important physical quantities. It is also used in the study of quantum systems, such as atoms and molecules, and in the development of quantum technologies.

5. Can the expectation value for a function with cusp be negative?

Yes, the expectation value for a function with cusp can be negative. This can occur when the function has a cusp at a point where the probability density function is relatively high. In this case, the negative expectation value indicates that there is a higher likelihood of finding the system in a state with a lower value of the physical quantity being measured.

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