Classically, if you put a particle in a box, it can have Zero kinetic energy (ie
ID: 478603 • Letter: C
Question
Classically, if you put a particle in a box, it can have Zero kinetic energy (ie the particle is not moving). But a quantum particle always has some kinetic energy. The difference between the lowest value of the potential energy and the energy of the lowest energy eigenstate is called the Zero Point Energy. Consider a 1D box that is one nanometer in length. What is the zero point energy for an electron, a proton, a methane molecule, and a gold atom in Joules? Which of these deviates the most from a classical particle?
Explanation / Answer
The energies which correspond with each of the permitted wavenumbers may be written as
En= n2h2/8mL2
n= the number of energy level
h =plancks constant
m= mass of particle
L = length of box
The zero-point energy is the lowest possible energy for the particle is found in state 1(n=1)
1. electron
m=9.1x10-31Kg
L=1nm=1x10-9m
E0= h2/8mL2=( 6.626 × 10 34 joulesecond)2/(8X9.1x10-31KgX[1x10-9m]2)
=6.030X10-21J
2. PROTON
m=1.626x10-27Kg
L=1nm=1x10-9m
E0= h2/8mL2=( 6.626 × 10 34 joulesecond)2/(8X1.626x10-27KgX[1x10-9m]2)
=3.37X10-23J
3.METHANE
m=2.66x10-26Kg
L=1nm=1x10-9m
E0= h2/8mL2=( 6.626 × 10 34 joulesecond)2/(8X2.66x10-26KgX[1x10-9m]2)
=2.06X10-24J
4.GOLD
m=3.27x10-25Kg
L=1nm=1x10-9m
E0= h2/8mL2=( 6.626 × 10 34 joulesecond)2/(8X3.27x10-25KgX[1x10-9m]2)
=1.67X10-25J