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It seems you're working on homework problems related to wavefunctions and probability densities. Here's an outline of the main tasks:

1. **Normalization of Wavefunctions:** For each wavefunction, you're asked to find the constant \( A \) to ensure that the total probability is 1.

2. **Expectation Values:** You need to compute the expectation values \( \langle x \rangle \) and \( \langle x^2 \rangle \) for the various wavefunctions.

3. **Standard Deviation:** Calculate the standard deviation \( \sigma \), using the expectation values.

Here are some key approaches:

### 1. **Normalization of the Wavefunction**
   You need to set up the integral of the probability density (squared wavefunction) over all space and set it equal to 1:
   \[
   \int_{-\infty}^{+\infty} |\psi(x)|^2 dx = 1
   \]
   This will allow you to solve for \( A \).

### 2. **Expectation Values**
   The expectation value of \( x \) is given by:
   \[
   \langle x \rangle = \int_{-\infty}^{+\infty} x |\psi(x)|^2 dx
   \]
   Similarly, for \( x^2 \):
   \[
   \langle x^2 \rangle = \int_{-\infty}^{+\infty} x^2 |\psi(x)|^2 dx
   \]

### 3. **Standard Deviation**
   The standard deviation is found using:
   \[
   \sigma = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}
   \]

Would you like help with a specific problem from the list, such as normalizing one of the wavefunctions, or calculating an expectation value?

Here are five questions to explore further:
1. How do you approach the normalization of a wavefunction with boundaries, such as \( \psi(x) \) defined over \( [0, a] \)?
2. How is the probability density related to physical measurements of the particle's position?
3. What is the importance of the standard deviation \( \sigma \) in quantum mechanics?
4. How would you calculate the expectation value of momentum, \( \langle p \rangle \), given the wavefunction \( \psi(x) \)?
5. How do boundary conditions affect the behavior of a wavefunction?

**Tip:** When solving for the normalization constant \( A \), remember to use symmetry if applicable—it can simplify the integrals!

For the wavefunction problems provided, including normalization, expectation values, and standard deviation.

Explore detailed solutions for wavefunction normalization, expectation values of x and x², and the calculation of standard deviation in quantum mechanics. Problems include various forms of ψ(x), with step-by-step guides. Suitable for undergraduate quantum mechanics students.

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