Probability of measuring the ground state of a particle (quantum mechanics)

In summary, to find the probability of measuring the ground-state energy of a particle in an infinite potential well with a wave-function \psi(x,0) = Ax(a-x), where a is the length of the well and A is the normalization constant, you can use the eigenvalues of the Hamiltonian in this potential, \phi_n=\sqrt{\frac{2}{a}}\sin(\frac{n\pi x}{a}), and take the absolute value squared of the inner product between \psi and \phi_1. This simplifies to \left[\int^a_0 \psi\phi_1 dx\right]^2, and after substituting in the value for A, you should get a
  • #1
BenR-999
5
0
Have to find the probability of measureing the ground-state energy of a particle.
-in infinite potential well 0<x< a
has wave-function [tex] \psi (x,0) = Ax(a-x) [/tex]
where a is the (known) length of the well, and the norm. const. A has already been found.

The eigenvalues of the hamiltonian in this potential are;
[tex] \phi_n = \sqrt{\frac{2}{a}} \sin(\frac{n \pi x}{a}}) [/tex]

I think that to do this i should take
[tex] \left| \langle \psi | \phi_n \rangle \right|^2 [/tex]
for n=1.

which becomes
[tex] \left[ \int^a_0 \psi \phi_1 dx \right] ^2 [/tex] (as all are real-valued)

I'm not sure if this is correct..it just seems a little to simple.

(with [tex] A=\frac{ \sqrt{30} }{a^{5/2} } [/tex]
i got [tex] 60/ \pi ^2 [/tex] which is obviously incorrect..as it is greater than 1.
But, if the method is correct and i have just made algebra error?

Thanks
 
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  • #2
Yes, your method is correct, and you just made an algebra mistake.
 
  • #3
beauty,
thanks mate
 

1. What is the ground state of a particle in quantum mechanics?

The ground state of a particle in quantum mechanics refers to the lowest energy state that a particle can occupy. It is also known as the "zero-point energy" state, as it is the state with the lowest possible energy.

2. How is the probability of measuring the ground state of a particle determined?

In quantum mechanics, the probability of measuring the ground state of a particle is determined by the wave function of the particle. The square of the wave function, known as the probability density, gives the likelihood of finding the particle in a specific state.

3. Can the ground state of a particle be measured with 100% certainty?

No, according to the principles of quantum mechanics, it is impossible to measure the ground state of a particle with 100% certainty. This is due to the inherent uncertainty in the position and momentum of a particle at the quantum level.

4. How does the concept of superposition affect the probability of measuring the ground state of a particle?

The concept of superposition, which states that a particle can exist in multiple states simultaneously, affects the probability of measuring the ground state of a particle. This is because the superposition of states can increase or decrease the probability of measuring a specific state, including the ground state.

5. Are there any real-world applications of understanding the probability of measuring the ground state of a particle?

Yes, understanding the probability of measuring the ground state of a particle is crucial in many technological advancements, such as quantum computing and quantum cryptography. It also plays a significant role in our understanding of atomic and molecular behavior in chemical reactions.

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