Probability of Finding a Particle | Quantum Science Philippines
Quantum Science Philippines
«
»

Probability of Finding a Particle

 

Prove that the total probability of finding a particle is conserved.

 

The norm of a state vector is given by:  \frac{d}{dt} \langle \psi (t) | \psi (t) \rangle.

In order to prove that the probability of finding a particle is conserved, we must obtain that

\frac{d}{dt} \langle \psi (t) | \psi (t) \rangle = constant

Now, we recall that from the 6th Postulate of Quantum Mechanics, the time evolution of the state vector | \psi (t) \rangle is governed by the Schrodinger Equation as,

i\hbar \frac{d}{dt} | \psi (t) \rangle = H(t)|\psi (t) \rangle                   [1]

\frac{d}{dt} | \psi (t) \rangle = \frac{1}{i \hbar} H(t)|\psi (t) \rangle               [2]

Taking the conjugate,

\frac{d}{dt} \langle \psi (t)| = -\frac{1}{i \hbar} \langle \psi (t)|H^\dagger (t)                [3]

but since H(t) by definition is Hermitian, that is, H^\dagger (t) = H(t), then

\frac{d}{dt} \langle \psi (t) | = - \frac{1}{i \hbar} \langle \psi (t) | H (t)                 [4]

Now, taking the scalar product of equations [4] and [2],

\frac{d}{dt} \langle \psi (t) | \psi (t) \rangle = \Big[ \frac{d}{dt} \langle \psi (t)| \Big] | \psi (t) \rangle + \langle \psi (t)| \Big[ \frac{d}{dt} | \psi (t) \rangle \Big]                 [5]

Substituting equations [2] and [4],

\frac{d}{dt} \langle \psi (t)| \psi (t) \rangle = - \frac{1}{i \hbar} \langle \psi (t)| H(t)| \psi (t) \rangle + \frac{1}{i \hbar} \langle \psi (t)| H(t)| \psi (t) \rangle    [6]

= \frac{1}{i \hbar} \langle \psi (t)| H(t) - H(t) | \psi (t) \rangle        [7]

= \frac{1}{i \hbar} \langle \psi (t)| 0 | \psi (t) \rangle                            [8]

\frac{d}{dt} \langle \psi (t) | \psi (t) \rangle = 0                                                                            [9] 

Hence, the total probability of finding a particle is conserved , since we obtained that

\mathbf{\frac{d}{dt} \langle \psi (t) | \psi (t) \rangle} = \mathbf{0}.    \Box

 

 

By: Joshua J. Ordeniza, MS Physics Student, MSU-IIT

 

 

This entry was posted under Quantum Physics, Quantum Science Philippines, Wavefunctions. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply