Probability Distribution Question

In summary, the stationary Schrodinger equation for a particle in a potential well has 2 solutions with different energy levels. At t = 0, the particle is in a superposition state of the two solutions. To calculate the probability distribution as a function of time, we need to consider time evolution in quantum mechanics. This can be done by making the time independent state into a time dependent one. More research and understanding of relevant equations and concepts is needed for further progress.
  • #1
Epideme
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0

Homework Statement


The stationary schrodinger equation for a particle moving in a potential well has 2 solutions
psi_1 (x) = e^(-ax^2), with Energy, E_1, and
psi_2 (x) = xe^(-ax^2) with Energy, E_2.

At t = 0 the particle is in the state
psi(x) = psi_1(x) + psi_2(x)

a)Calculate the probability distribution for the particle as a function of time
b)Find the time at which the probability distribution returns to the initial value


Homework Equations





The Attempt at a Solution


Again no progress on this one, i have no idea, thank you in advance for any help offered.
 
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  • #2
of course one always have ideas, come on, you can atleast try to find the equations which you think are relevant and say what you DO think you understand and what you dont't. Just saying "i have no idea" will not help you. You HAVE to give attempt to solution if you want help here.

Hint: Look for time evolution, how is time evolution implemented in quantum mechanics? Look through your books and notes. How do we make a time independent state to become time dependent?
 

1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes occurring in a random experiment. It can be represented in various forms, such as a table, graph, or equation.

2. What are the types of probability distributions?

The types of probability distributions include discrete distributions, which describe outcomes that can only take on certain values, and continuous distributions, which describe outcomes that can take on any value within a certain range. Examples of discrete distributions include the binomial and Poisson distributions, while examples of continuous distributions include the normal and exponential distributions.

3. How is a probability distribution different from a probability density function?

A probability distribution describes the likelihood of different outcomes occurring in a random experiment, while a probability density function (PDF) describes the relative likelihood of different outcomes occurring within a specific range. The area under a PDF curve represents the probability of an outcome occurring within that range.

4. What is the mean and standard deviation of a probability distribution?

The mean of a probability distribution is the average value of all possible outcomes and is denoted by µ. The standard deviation is a measure of the spread of the distribution and is denoted by σ. It represents the average amount that each outcome deviates from the mean.

5. How is a probability distribution used in real-life applications?

Probability distributions are used in various real-life applications, such as risk assessment, finance, and statistical analysis. For example, in risk assessment, probability distributions are used to calculate the likelihood of certain events occurring, such as natural disasters. In finance, probability distributions are used to model stock prices and assess risk in investments. In statistical analysis, probability distributions are used to make predictions and draw conclusions from data.

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