Brushing Up: Understanding Why We Must Do |$\psi_i$|$^2$

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In summary: So basically if you have a vector with components that represent a state, and you want to know the probability of being in that state, you take the square of the amplitude of the vector (in the complex number sense), and that is what you get.
  • #1
FrogPad
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I am brushing up on some basic quantum mechanics that we covered in the review for a course I just started. For some reason I cannot remember why the following is the case.

So let's say we have [tex] \psi [/tex] representing a vector with components that represent a state.

Why do we have to do: [tex] | \psi_i |^2 [/tex] to get the probability that we are in that given state?

I know I HAVE to, but I cannot remember WHY.

thanks
 
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  • #2
Because you are taking the [tex] \psi_i [/tex] to be orthogonal and [tex] \psi [/tex] to be normalized. So the probability to be in a given state is <psi_i|psi> which is what you said.
 
  • #3
Dick said:
Because you are taking the [tex] \psi_i [/tex] to be orthogonal and [tex] \psi [/tex] to be normalized. So the probability to be in a given state is <psi_i|psi> which is what you said.

Would you mind expanding upon what you said? I cannot fully follow.

I have unfortunately never taken a proper quantum class. The two courses that I have even taken are a modern physics course (sophomore level -- talks about slit experiments and the likes), and a properties of solids (junior level - energy bands, probability amplitudes, etc...)

The course I am taken is a solid state device physics course. The review (first day) started with bra-ket notation; this is something I have never encountered before. Trying to review (more proper, catch up) has led me to a more thorough reading of the basics (including review of linear algebra).
 
  • #4
If you know what a probability amplitude is then you are basically there. If you know the amplitude then you square it (in the complex number sense) to get the probability. If psi is split into a sum of orthogonal (so <psi_i|psi_j>=0 if i is not equal to j) components psi=sum(psi_i) and is normalized, so <psi|psi>=1. Then the probability of being in the state i is <psi_i|psi>=<psi_i|psi_i>. I don't think I'm explaining this very well, because it's really close to being one of the assumptions of quantum mechanics. Maybe somebody else can clarify.
 
  • #5
This has to do with the amplitude as Dick said, if you look up the word probablity amplitude in your book om statistics, you will se parallells =)

I remember i struggeled a lot on this too in the begining, but then I compared that to the stuff I learned i statistics, and things became clearer.
 

1. What is |$\psi_i$|$^2$ and why is it important in brushing up?

|$\psi_i$|$^2$ refers to the probability of finding a particle in a specific state or location. In brushing up, it is important because it helps us understand the likelihood of finding a particle in a certain state, which is crucial in many scientific experiments and applications.

2. How is |$\psi_i$|$^2$ related to the Schrödinger equation?

|$\psi_i$|$^2$ is directly related to the Schrödinger equation, which is a fundamental equation in quantum mechanics. The square of the wave function (|$\psi_i$|$^2$) is used to calculate the probability density of a particle described by the wave function, as determined by the Schrödinger equation.

3. Can you explain the concept of superposition and how it relates to |$\psi_i$|$^2$?

Superposition refers to the ability of quantum particles to exist in multiple states at once. This is described by the wave function, which is the superposition of all possible states. The square of the wave function (|$\psi_i$|$^2$) represents the probability of finding a particle in a particular state, thus showing the concept of superposition in action.

4. How does the concept of |$\psi_i$|$^2$ help us understand the behavior of particles in quantum systems?

|$\psi_i$|$^2$ is a crucial concept in quantum mechanics that helps us understand the behavior of particles in quantum systems. It allows us to calculate the probability of finding a particle in a specific state, which is essential in predicting and understanding the behavior of particles at the quantum level.

5. Are there any practical applications of understanding |$\psi_i$|$^2$?

Yes, there are many practical applications of understanding |$\psi_i$|$^2$. For example, it is used in the design of electronic devices, pharmaceuticals, and materials. It is also important in fields such as quantum computing and quantum cryptography. Understanding |$\psi_i$|$^2$ is crucial in advancing our understanding and application of quantum mechanics in various industries.

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