Expected value inequality

In summary, we are trying to understand why the expectation value of the momentum squared, given by p=-i\hbar{d\over dx}, is always greater than zero for all normalized wavefunctions. We can argue that this is true due to the Hermitian nature of p, but the strict inequality is still unclear. It is possible that this would be trivial otherwise. Additionally, if there exists a wavefunction \psi with a zero inner product with p^2, then it must have a zero norm and may not be normalizable.
  • #1
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Homework Statement


Why is [itex]\langle p^2\rangle >0[/itex] where [itex]p=-i\hbar{d\over dx}[/itex], (noting the ***strict*** inequality) for all normalized wavefunctions? I would have argued that because we can't have [itex]\psi=[/itex]constant, but then I thought that we can normalize such a wavefunction by using periodic boundary conditions... So I don't how to argue that the inequality should be strict... Is it that otherwise it would be trivial?


Homework Equations


[itex]p=-i\hbar{d\over dx}[/itex]


The Attempt at a Solution


clearly, [itex]\langle \psi|p^2|\psi\rangle = \langle p\psi|p\psi\rangle \geq 0[/itex] since p is Hermitian. But why the strict inequality??
 
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  • #2
When would that inner product with zero? and what would happen then? ;)
 
  • #3
Well, if there exists [itex] \psi \in D(p) [/itex], so that [itex] \langle \psi,p^2 \psi\rangle [/itex] = 0, then [itex] ||p\psi|| [/itex] has 0 norm. Which vector has 0 norm ? Is it normalizable ?
 

What is the concept of expected value inequality?

The expected value inequality is a mathematical concept that is used to compare the expected values of two random variables. It states that if two random variables have the same probability distribution, then the one with the higher expected value is more likely to produce a larger outcome.

How is expected value inequality used in decision making?

Expected value inequality is used in decision making to determine the most favorable option among multiple choices. By calculating the expected value for each option, one can choose the option with the highest expected value, which is most likely to result in a larger outcome.

What is the relationship between expected value inequality and risk?

The relationship between expected value inequality and risk is that the higher the expected value, the lower the risk. This is because a higher expected value indicates a higher likelihood of a larger outcome, reducing the risk of a smaller outcome.

How is expected value inequality different from expected value?

Expected value inequality compares the expected values of two random variables, while expected value calculates the average of a single random variable. Expected value inequality is used to compare options, while expected value is used to estimate the long-term outcome of a random variable.

What factors can affect expected value inequality?

The factors that can affect expected value inequality include the probability distribution of the random variables, the potential outcomes, and the decision maker's attitude towards risk. Changes in these factors can alter the expected values and impact the comparison between options.

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