Problem[]
This problem is a step-by-step derivation of the famous uncertainty principle in quantum mechanics.
Part 1: Let
and
, show that
- .
Hint:
Part 2: Show (by calculation) that
and
- .
Then show that
- .
Part 3: It is proven in this problem that , show that .
Solution[]
Part 1
The standard deviation of position is
and the standard deviation of momentum is
- .
Now this means,
- .
The Cauchy-Schwarz inequality states , where .
Therefore, .
Since ,
replacing with and with yields
- .
Part 2
Taking the complex conjugate yields
- .
This means
.
Subsequently,
- .
Part 3
Since ,
- .