# How to solve for D and sketch the wavefunctions

• vbrasic
In summary, the conversation discusses solving a potential well problem using the wavefunction inside and outside the well, and using the continuity and normalization conditions to determine coefficients. The conversation also mentions using these results to sketch the wavefunctions and determine energy levels, probability density, and expectation values for the particle's position.
vbrasic

## Homework Statement

See attached image. The potential in question is, ##-V_0## for ##0<r<a,## and ##0## for ##r\geq a.##

## Homework Equations

$$\sinh(x)=\frac{e^x+e^{-x}}{2}$$
$$\cosh(x)=\frac{e^x-e^{-x}}{2}$$

## The Attempt at a Solution

I know that the wavefunction for ##r<a## is given by ##Asin(k_1r),## where ##k_1## is $$\frac{\sqrt{(2m(E+V_0)}}{\hbar},$$ and that the wavefunction for ##r\geq a## is $$De^{-k_2r},$$ where $$k_2=\frac{\sqrt{-2mE}}{\hbar}.$$ I can write ##De^{-k_2r}## as ##D(\sinh(k_2r)-\cosh(k_2r)).## Then imposing continuity, I have the system, $$A\sin(k_1a)=D(\sinh(k_2a)-\cosh(k_2a))$$ $$k_1A\cos(k_1a)=k_2D(\cosh(k_2a)-\sinh(k_2a)),$$ such that ##k_1 \cot(k_1a)=-k_2.## Then imposing normalization, on inner wavefunction, as per the text's suggestion, I get ##A=\frac{1}{k_1},## so that $$D=\frac{\frac{1}{k_1}\sin(k_1a)}{\sinh(k_2a)-\cosh(k_2a)}.$$ Naturally, we can rewrite ##k_2## in terms of ##k_1## from the continuity relation, from which we can sketch the wavefunctions. I'm not entirely sure if what I've done so far is correct, and if so how I would even go about sketching these wavefunctions, and doing parts b), c), and d).

For part b), I'm guessing that we can just equate, $$-\frac{-\sqrt{-2mE}}{\hbar}=k_2$$ and rearrange for ##E.## Though I still have no idea how to semi-quantitatively sketch such a function. As well, by the logic that the probability of the particle to be somewhere in all space, the wavefunction outside the well must decay. (That should describe the behavior outside the well.)

<Moderator's note: formatting fixed. Please use ## ## for inlined LaTeX.>

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Your approach to solving this problem is correct. You have correctly identified the wavefunctions inside and outside the potential well, and have used the continuity and normalization conditions to determine the coefficients A and D.

To sketch the wavefunctions, you can plot the functions ##Asin(k_1r)## and ##De^{-k_2r}## separately, and then combine them to get the overall wavefunction. The wavefunction inside the potential well will have a sinusoidal shape, while the wavefunction outside the potential well will decay exponentially. You can use the continuity condition to determine the values of k1 and k2, and then plot the wavefunction for different values of V0.

For part b), you are correct in assuming that the energy can be determined by equating ##-\frac{-\sqrt{-2mE}}{\hbar}=k_2## and solving for E. To sketch the energy levels, you can plot the energy values for different values of V0 and observe how they change with varying potential depth.

For part c), you can use the wavefunction to calculate the probability density of finding the particle at a particular point. This can be done by taking the square of the absolute value of the wavefunction at that point. You can then plot the probability density for different values of V0 to see how it varies.

For part d), you can use the wavefunction to calculate the expectation value of the position of the particle. This can be done by multiplying the wavefunction by the position operator and integrating over all space. You can then plot the expectation value for different values of V0 to see how it changes.

I hope this helps. Good luck with your calculations.

## 1. What is the purpose of solving for D and sketching wavefunctions?

Solving for D and sketching wavefunctions allows us to visualize and understand the behavior of a wave in a given system. It also helps us to calculate important properties of the wave, such as its energy and momentum.

## 2. How do I solve for D in a wavefunction?

To solve for D in a wavefunction, you will need to use the Schrödinger equation and apply the appropriate boundary conditions for the specific system you are studying. This will typically involve using mathematical techniques such as integration and differential equations.

## 3. What information can be obtained from a wavefunction sketch?

A wavefunction sketch can provide information about the shape, amplitude, and wavelength of the wave. It can also show the probability distribution of the wave in a given system.

## 4. Can D and the wavefunction be solved for any type of wave?

Yes, D and the wavefunction can be solved for any type of wave, including electromagnetic waves, sound waves, and quantum mechanical waves. However, the specific techniques and equations used may vary depending on the type of wave.

## 5. How does solving for D and sketching wavefunctions relate to real-world applications?

Solving for D and sketching wavefunctions is essential in understanding and predicting the behavior of waves in various systems, such as electronic circuits, optical devices, and chemical reactions. It is also crucial in the field of quantum mechanics, which has numerous technological applications, such as quantum computing and cryptography.

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