Probability of being in a state is given, Find the normalised wavefunction

• dipanshum
In summary, the probability of finding the particle in state ψ1 is given in the question but it is necessary to compute the probability using the normalized wavefunction ψ = 1/3 ψ1 + 2√2/3 ψ2. To do this, one must calculate the values of α and β in the equation ψ = αψ1 + βψ2, where α2 + β2 = 1 after normalization. The probability of the state being ψ2 can be found by substituting the values of α and β in the normalized wavefunction ψ=1/√3 ψ1 + √(2/3)ψ 2.
dipanshum
Homework Statement
A particle can be in two different states given by orthonormal wavefunctions ψ1 and ψ2. If the probability of being in state ψ1 is 1/3, find out normalized wave function for the particle.
Relevant Equations
ρ(r,t) = |ψ(r,t)|^2
where ρ= probability
Should I treat ψ1 as ψ and ψ 2 as ψ*?

What you have written down is not the probability of finding the particle in a particular state, it is the probability density of finding the particle at position ##r##. That the particle is in a state where it is in either ##\psi_1## or ##\psi_2## means that it is a linear combination of those states, i.e., ##\psi = \alpha \psi_1 + \beta \psi_2##. You need to find ##\alpha## and ##\beta##. (Note that there is an inherent phase ambiguity.)

dipanshum
Orodruin said:
What you have written down is not the probability of finding the particle in a particular state, it is the probability density of finding the particle at position ##r##. That the particle is in a state where it is in either ##\psi_1## or ##\psi_2## means that it is a linear combination of those states, i.e., ##\psi = \alpha \psi_1 + \beta \psi_2##. You need to find ##\alpha## and ##\beta##. (Note that there is an inherent phase ambiguity.)

Ok, I worked out the sum. Please tell me if I'm right.
The normalized wavefunction is:
ψ = 1/3 ψ1 + 2√2/3 ψ2

How do you compute the probability of being in the state ##\psi_1##?

Orodruin said:
How do you compute the probability of being in the state ##\psi_1##?
The probability of ψ1 is given in the question.

dipanshum said:
The probability of ψ1 is given in the question.

Can you check that your ##\psi## has this probability? You said:

dipanshum said:
The normalized wavefunction is:
ψ = 1/3 ψ1 + 2√2/3 ψ2

What are the respective probabilities of finding the system in ##\psi_1## and ##\psi_2## using your ##\psi##?

dipanshum said:
The probability of ψ1 is given in the question.
Yes, but your job is to show that your state satisfies that and in order to do that you must compute the probability of your particle being in the state ##\psi_1##.

PeroK said:
Can you check that your ##\psi## has this probability? You said:
What are the respective probabilities of finding the system in ##\psi_1## and ##\psi_2## using your ##\psi##?
no only probability of ψ is given, but that is sufficient data. and what you suggested already helped me and the answer is right.
what happened is after normalization one will get α22=1
then calculate and put values of α, β in the equation ψ=αψ1+βψ2

dipanshum said:
no only probability of ψ is given, but that is sufficient data. and what you suggested already helped me and the answer is right.
what happened is after normalization one will get α22=1
then calculate and put values of α, β in the equation ψ=αψ1+βψ2

What's the probability of the state being ##\psi_2##?

dipanshum said:
It is not.

Orodruin said:
It is not.
Okay so, yes Orodruin you are right, and I was wrong.
The final normalised function would be ψ=1/√3 ψ1 + √(2/3)ψ 2.
Tell me if I'm right this time.

1. What is the probability of being in a specific state?

The probability of being in a specific state is given by the square of the magnitude of the wavefunction at that state. This can be calculated by taking the absolute value of the wavefunction and squaring it.

2. How is the wavefunction normalised?

The wavefunction is normalised by dividing it by the square root of the integral of the absolute value of the wavefunction squared over all possible states. This ensures that the total probability of being in any state is equal to 1.

3. What is the significance of the normalised wavefunction?

The normalised wavefunction represents the probability amplitude of finding a particle in a specific state. It is a fundamental concept in quantum mechanics and is used to calculate the probability of a particle's behavior or location.

4. Can the wavefunction be normalised to a value other than 1?

No, the wavefunction must always be normalised to a value of 1 in order to accurately represent the probability of a particle's behavior. If the wavefunction is not normalised, it would not accurately reflect the probability of finding a particle in a specific state.

5. How is the normalised wavefunction used in quantum mechanics?

The normalised wavefunction is used to calculate the probability of a particle's behavior or location in quantum mechanics. It is also used in calculations to determine the energy levels and properties of quantum systems.

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