8 27.1 The Heisenberg uncertainty principles Example 4: Minimum energy of a particle in a box -zero point energy Solution: For a particle in a box of length L (4p)2=(p-p) (p2-2p-p2) P-p=p h (4p)2=p2≥( The microscopic In particle cannot E ≠0 be rest! 2m 2mL s 27.1 The Heisenberg uncertainty principles Example 5: a proton is known to be in the nucleus of an atom. The size of the nucleus is about 1.0x10- 4m. Find a Pmin=?; b vmin=? For nonrelativistic situation c is it reasonable to express its momentum in classical expression? Solution: a. According to the uncertainty principle h6.626×10 4p 10×104=6.6×10-20kgm/s b The minimum speed of the proton 6.6×10 40×107m/s 1.67×10 6
6 Example 4: Minimum energy of a particle in a box —zero point energy 2 2 2 2 2 2 2 ( 2 ) ( ) ( ) p p p p pp p p p p av av = − = = − − ∆ = − 0 2 2 ( ) ( ) 2 2 2 2 2 2 = ≥ ≠ = ≥ mL h m p E x h p p ∆ ∆ For a particle in a box of length L §27.1 The Heisenberg uncertainty principles Solution: The microscopic particle cannot be rest! Example 5: A proton is known to be in the nucleus of an atom. The size of the nucleus is about 1.0×10-14 m. Find a. pmin=?; b. vmin=? For nonrelativeistic situation. c. is it reasonable to express its momentum in classical expression? Solution: a. According to the uncertainty principle 6.6 10 kg m/s 1.0 10 6.626 10 20 14 34 = × ⋅ × × = = − − − x h p ∆ ∆ b. The minimum speed of the proton 4.0 10 m/s 1.67 10 6.6 10 7 27 20 min = × × × = = − − m p v §27.1 The Heisenberg uncertainty principles
8 27.1 The Heisenberg uncertainty principles c. The fraction of the speed of light 4.0×10 =0.13 c3.0×10 y ≈1.0 It is reasonable to express its momentum in classical expression. s 27.1 The Heisenberg uncertainty principles >Implications of the energy-time uncertainty principle a. The mass of fundamental particles According to special relativity Em=mc2→AEt=c2m According to the energy-time uncertainty principle c2AmAt≥h The mean lifetime of a free neutron is 888 s then 6.626×10 =829×10-k c2r(3×105)2(88》
7 c. The fraction of the speed of light 1.0 1 / 1 0.13 3.0 10 4.0 10 2 2 8 7 ≈ − = = × × = v c c v γ It is reasonable to express its momentum in classical expression. §27.1 The Heisenberg uncertainty principles ¾Implications of the energy-time uncertainty principle a. The mass of fundamental particles According to special relativity E mc ∆E c ∆m 2 rest 2 rest = ⇒ = According to the energy-time uncertainty principle c ∆m∆t ≥ h 2 8.29 10 kg (3 10 ) (888) 6.626 10 54 8 2 34 2 − − = × × × = = c t h m ∆ ∆ The mean lifetime of a free neutron is 888 s, then §27.1 The Heisenberg uncertainty principles