Probability of finding a particle in a certain state, using projection

In summary, The method discussed uses projection to calculate the probability of finding a particle in a certain state. The process involves defining an operator and using it to find the probability of measurement for a specific property. An example is given to explain this process, but the calculation of the projection operator is still unclear.
  • #1
dreamspy
41
2
I was reading about a certain methood that uses projection to calculate the probability of finding a particle in a certain state. The explanation is not detailed enough for me to get my head around how to use it, but maybe some of you people are familiar with the methood? The methood goes like this:

We have that [tex]\epsilon \subset H[/tex] is a subspace with an orthonormal basis [tex]\{\phi_j\}[/tex] (here H is Hilbert space). We define an operator [tex]\Pi:H\rightarrow \epsilon[/tex] like this:

[tex]\Pi\psi=\sum_j<\phi_j,\psi>\phi_j[/tex]

Let's say that [tex]A[/tex] is a measurable property of a particle, and [tex]\hat A[/tex] is the corresponding operator. Let's make [tex]a[/tex] to be one of the eigenvalues of [tex]\hat A[/tex] and [tex]\epsilon_a[/tex] is the corrisponding eigenspace. That is: [tex]\epsilon_a [/tex] is the subspace in H that is spanned by all eigenfunctions of [tex]\hat A[/tex] with eigenvalue [tex]a[/tex]. If the particle has the wavefunction [tex]\psi_t[/tex] then the probability to measurement of [tex]A[/tex] will give the result [tex]a[/tex] is:

[tex]P(A=a)=\parallel\Pi_a\psi_t\parallel^2[/tex]

Now there is given an example to explain this. We are given wave function

[tex]\Psi=(A+Bx)e^{-x^2/a^2}[/tex]

And the parity operator [tex]\hat p[/tex] which functions like this:

[tex]\hat p \Psi(x) = \Psi(-x)[/tex]

Now it's easy to show that the eigenvalues of the parity operator are [tex]\pm 1[/tex] and the corresponding eigenfunctions are:
all even functions for eigenvalue +1
all odd functions for eigenvalue -1

Now the probability of getting [tex]p = +1[/tex] is given by:

[tex]P(p=+1) = \parallel \hat\Pi_{+1}\Psi\parallel^2 = \parallel Ae^{-x^2/a^2}\parallel^2[/tex]

Now what I don't get is how [tex]\hat \Pi_{+1}\Psi[/tex] was calculated. Anyone care to shed a light on this for me?
 
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  • #2
The projection operator is defined to give zero when applied to an odd function and to give the function back when it is applied to an even function. So

[tex] \Pi_{+1} ( xe^{-x^2/a^2} )= 0 [/tex]

and

[tex] \Pi_{+1} ( e^{-x^2/a^2} )= e^{-x^2/a^2} [/tex]
 
  • #3
But what about the sum:

[tex]\Pi\psi=\sum_j<\phi_j,\psi>\phi_j[/tex]

I'm not sure how I would use this in general with other problems.
 
  • #4
I think I got it now, but I still find the definition of [tex]\Pi[/tex] a little bit odd.
 
  • #5
dreamspy said:
I think I got it now, but I still find the definition of [tex]\Pi[/tex] a little bit odd.

I doubt you are the only one.

[tex]\p[/tex]
 

Related to Probability of finding a particle in a certain state, using projection

1. What is the concept of "probability of finding a particle in a certain state"?

The probability of finding a particle in a certain state refers to the likelihood of observing a particle in a specific quantum state during an experiment or measurement. This probability is described by quantum mechanics and can be calculated using mathematical equations.

2. How is the probability of finding a particle in a certain state determined?

The probability of finding a particle in a certain state is determined by taking the square of the absolute value of the projection of the particle's state vector onto the state being observed. This is known as the Born rule and is a fundamental principle in quantum mechanics.

3. Can the probability of finding a particle in a certain state be greater than 1?

No, the probability of finding a particle in a certain state cannot be greater than 1. This is because the probability is calculated as the square of the state vector projection, which can never be greater than 1.

4. How does the projection operator affect the probability of finding a particle in a certain state?

The projection operator is used to determine the probability of finding a particle in a certain state by projecting the state vector onto the observed state. This allows us to calculate the probability using the Born rule and provides a mathematical representation of the probability of finding a particle in a certain state.

5. Why is the concept of probability important in quantum mechanics?

The concept of probability is important in quantum mechanics because it helps us understand and predict the behavior of subatomic particles. In quantum mechanics, the behavior of particles is described by probability distributions rather than definite outcomes, and the measurement of a particle can change its state. Probability allows us to make predictions and understand the probabilistic nature of the quantum world.

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