The Heisenberg uncertainty principle demonstrate the symmetry between the particles position and momentum. This can further be extended to the particles probability amplitudes in position (x, t) and momentum (p, t). If 4(x, t) = √2 dp (p, t)e ³x is the Fourier expansion of the probability amplitude for (p,t) = x (x, t)e-³x is its Fourier transform, prove that = + (p, t)| ²dp. position and √√2h +x (x, t) |² dx = +00 -+∞0

The Heisenberg uncertainty principle demonstrate the symmetry between the particles position and momentum. This can further be extended to the particles probability amplitudes in position (x, t) and momentum (p, t). If 4(x, t) = √2 dp (p, t)e ³x is the Fourier expansion of the probability amplitude for (p,t) = x (x, t)e-³x is its Fourier transform, prove that = + (p, t)| ²dp. position and √√2h +x (x, t) |² dx = +00 -+∞0