Chapter 6. Symmetries and Conservation Laws - (3)

6.3 Conservation Laws

In Classical mechanics, the meaning of a conservation law is straightforward. The quantity of energy is same before and after some event.

However, in Quantum mechanics, we need thorough conversations.
In Quantum mechanics, conservation of an observable Q means 

1. <Q> is independent of time

2. The probability of getting any particular valude is independent of time

There are two 

Let the observable in question does not depend explicitly on time : $$\frac{\partial Q}{\partial t}$$

 

Then the generalized Ehrenfest theorem tells us that the expectation value of Q is independent of time if the $$\hat{Q}$$ commutes with hamiltonian.

 

$$Q\to q_n$$

$$P(q_n)=|<f_n|\psi (t)>{|}^2$$ 

$$f_{n}standsforeigenfunction.$$ 

 

$$\hat{Q}|f_n>=|q_n{f}_n>$$

 

$$\psi (t)>=\sum _m{e}^{\frac{-i{E}_{n}t}{\hslash }}{C}_m|{\psi }_m>$$

So we have $$\hat{H}|{\psi }_m>=E_m|{\psi }_m>$$ ($$E_m$$ are eigenvalues for each wave eqs.

And we rewrite $$P(q_n)=|<f_n|\psi (t)>{|}^2$$ into 

$$P(q_n)=|\sum _m{e}^{\frac{-i{E}_{n}t}{\hslash }}{C}_m<f_n|{\psi }_m(t)>{|}^2=|e^{\frac{-i{E}_{n}t}{\hslash }}{C}_m{|}^2=|C_n{|}^2$$

 

Since $$[\hat{H},\hat{Q}]=0$$, we can find common eigenstates and they form complete set. We have $$<f_n|{\psi }_m>=<{\psi }_n|{\psi }_m>={\delta }_{nm}$$

 

Reference : Introduction to Quantum Mechanics 3rd ed. (David J. Griffith)