5/13 Potential energy diagram and potential well

Potential energy diagram
1&2.The range of motion is from -5cm to 5cm because the energy the particle is bigger than well in this region.
3. The more time the particle spends in one region, the more likely it is to be detected in that region. The particle spends more time to the left of zero because its kinetic energy (and hence its speed) is much smaller in that region. Therefore, the particle is much more likely to be detected to the left of zero.
4. The turning points move outward from the origin by a factor of the square root of two because 1/2 kx^2 = U
5. The shape of the kinetic energy is a parabola, with the opending down.
6. At the turning point. Because the kinetic energy of the particle at the turning points are zero, it is easier to be detected.

 

Potential well:

1. E = n² h² / 8 m L².

Evaluating this expression yields:
E = (1)² (6.626 x 10^-34 J s)² / 8 (1.673 x 10^-27 kg) (10 x 10^-15 m)²
E = 3.3 x 10^-13 J
E = 2.1 MeV

2. Since n = 2, and E = n² h² / 8 m L²
the energy of the first excited state for an infinite well is
E = 4 (2.1 MeV)
E = 8.4 MeV
But E = 8.4 MeV is not an allowed energy level in the finite well.

3. Since the wavefunction can penetrate into the "forbidden" regions, the wavelength of the wavefunction is larger in the finite well than in a same width infinite well. A larger wavelength implies a lower energy.



 

4. When the depth of the potential well is decreased from 50 MeV to 25 MeV, the n = 3 state becomes nearer in energy to the "top" of the well. As the energy level gets closer to the "top" of the well, the penetration depth increases, leading to a longer and longer wavelength for the wavefunctio, and a longer wavelength implies a smaller energy.

5. The penatrate depth will decrease when the mass of the particle is increased because macro-particle cannot penentrate through the forbidden region.