Wavefunction normalisation and expectation values

• MoAli
In summary, the conversation discusses a homework problem involving finding expectation values for position and momentum without using integrals. The solution involves using complex numbers and being careful with integration. The individual is also still trying to learn latex and prefers attaching images instead of writing out equations on a forum's keyboard.
MoAli

Homework Statement

See Image, Sorry Its easier for me to attach images than writing all equation on the forum's keyboard!
I only need to check if I'm working it out correctly up to the position expectation value because I don't want to dive in the rest on wrong basis !

The Attempt at a Solution

You need to try some latex. What you have is correct, except that right at the end in the expected value of ##x##, ##\alpha## leapt from the denominator to the numerator.

##\langle x \rangle = \frac{12}{13 \alpha}##

Yeah still trying to learn latex, anyway, where the question asks to prove momentum expectation value is zero without integrals I get stuck, I got <p^2> = \frac{12/h^2}{13 } which still doesn't make sense to me, the units don't work!

MoAli said:
Yeah still trying to learn latex, anyway, where the question asks to prove momentum expectation value is zero without integrals I get stuck, I got the \langle p^2 \rangle to be \frac{12h^2}{72}

Regarding ##\langle p \rangle## you might like to think about complex numbers and expectation values.

I haven't tried to calculate ##\langle p^2 \rangle## or ##\langle x^2 \rangle##. You just need to be careful with the integration.

1. What is wavefunction normalisation?

Wavefunction normalisation refers to the process of ensuring that the total probability of finding a particle within a given region is equal to 1. This is achieved by normalising the wavefunction, which is a mathematical function that describes the quantum state of a particle.

2. Why is wavefunction normalisation important?

Wavefunction normalisation is important because it ensures that the probabilities calculated from the wavefunction are meaningful and consistent with the principles of quantum mechanics. It also allows us to interpret the wavefunction as a probability distribution, which is essential for understanding the behaviour of particles at the quantum level.

3. What is an expectation value?

An expectation value is the average value of a physical quantity that we expect to measure in a given quantum state. It is calculated by taking the weighted average of all the possible values of the quantity, where the weights are determined by the probability of measuring each value.

4. How is the expectation value related to the wavefunction?

The expectation value of a physical quantity is related to the wavefunction through the mathematical operation of taking the inner product of the wavefunction with the operator corresponding to the physical quantity. This operation yields a complex number, and the square of its magnitude represents the expectation value.

5. Can the expectation value be negative?

Yes, the expectation value can be negative. This occurs when the wavefunction has both positive and negative values, and the operator corresponding to the physical quantity has a mix of positive and negative eigenvalues. In this case, the positive and negative contributions to the expectation value may cancel each other out, resulting in a negative value.

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