Question 41.5: A Ground-State Energy Free Particle Take the limit of E1 = h......

Physics for Scientists and Engineers Foundations and Connections [987819]

A Ground-State Energy Free Particle

Take the limit of $E_1 = h^2 /8mL^2$ (Eq. 41.16 with n = 1) to find the zero-point energy of a free particle.

E_n=\left(\frac{h^2}{8 \pi^2 m}\right)\left(\frac{n^2 \pi^2}{L^2}\right)=n^2\left(\frac{h^2}{8 m L^2}\right) \quad \quad (41.16)

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

INTERPRET and ANTICIPATE
The key to this problem is knowing what limit we must take in order to go from the energy expression for a trapped particle to that for a free particle. There are only two parameters we can change: m and L. Changing the mass does not set a particle free, but increasing the width of the well does. An infinitely wide well means that the particle can go anywhere.

SOLVE
Take the limit as L → ∞.

\underset{L\rightarrow \infty }{lim} E_1=\underset{L\rightarrow \infty }{lim} \frac{h^2}{8 m L^2} \rightarrow 0

CHECK and THINK
When a particle is free (not part of any system), it can have any energy value. So its minimum energy is zero and it may be at rest. Only a confined particle is required to have nonzero energy and to be in motion.

Related Answered Questions

Question: 41.11

Hawking Radiation Is NOT Science Fiction In the previous example, we imagined a whole meal being created from nothing. The result was that such a meal could only exist for a negligible amount of time. But don’t think that such events are impossible. The British physicist Stephen Hawking has a ...

Verified Answer:

INTERPRET and ANTICIPATE Like the previous example...

Question: 41.3

Average Position for a Particle in an Infinite One-Dimensional Well, Classical Approach Consider the hockey puck trapped between the walls of the one-dimensional box in Figure 41.1. As described in Section 41-2, when the puck encounters a wall, it reverses direction instantaneously and there are no ...

Verified Answer:

INTERPRET and ANTICIPATE The expectation value is ...

Question: 41.10

Science Fiction As we saw in Chapter 39’s CASE STUDY (page 1270), science fiction writers often use real physics in their stories. Let’s test an idea a writer had. Feeding people on deep space missions is challenging. In a sense, the mission would need to somehow pack all the nutrition the crew ...

Verified Answer:

INTERPRET and ANTICIPATE The key to this problem i...

Question: 41.9

Modeling Iodine A number of times throughout this book, we have modeled molecular bonds as tiny springs connecting atoms. (For examples, see Figures 5.15, 14.21, and 21.25.) In particular, a vibrating diatomic molecule may be modeled as a simple harmonic oscillator (Fig. 41.21). In this problem, we ...

Verified Answer:

INTERPRET and ANTICIPATE Modeling the iodine molec...

Question: 41.8

In Which State Is the Oscillator? A simple harmonic oscillator in its ground state has a frequency of 3.450 MHz. Now its energy is found to be 2.400 × 10^-26 J. What state is it in? ...

Verified Answer:

INTERPRET and ANTICIPATE The frequency of the grou...

Question: 41.7

CASE STUDY The Probability of Proton Tunneling in the Sun Equation 41.22 is for a rectangular barrier. Although the barrier in Figure 41.18 is not rectangular, use Equation 41.22 to make an order-of-magnitude estimate of the probability of tunneling required for two protons to fuse together in the ...

Verified Answer:

INTERPRET and ANTICIPATE We expect to find a small...

Question: 41.6

The Penetration Distance of Particle in a Finite Square Well When a particle in a finite square well is in its ground state, its penetration distance is Λ1. The well’s height is U0 = 5.55 eV. When the same system is in an excited state with an energy of 5.35 eV, its penetration distance is 4Λ1. ...

Verified Answer:

INTERPRET and ANTICIPATE The penetration distance ...

Question: 41.4

Average Position for a Particle in an Infinite One-Dimensional Well, Quantum Approach Find the expectation value of x for a particle in an infinite well of width L. This time apply the solution we found to Schrödinger’s equation for this situation. ...

Verified Answer:

INTERPRET and ANTICIPATE This problem is the same ...

Question: 41.2

Throwing Dice A die is a cube whose six sides are marked by dots; one side has one dot, another two dots and so forth. If you roll a single die, the probability of rolling a 4 is 1/6 because there are six sides and only one has exactly four dots. If you roll two dice simultaneously, what is the ...

Verified Answer:

INTERPRET and ANTICIPATE This is a classical mecha...

Question: 41.1

A Hockey Puck in a Box Suppose that a hockey puck’s mass is 0.160 kg and it is in a box of length 6.35 m (Fig. 41.1). Find the system’s zero-point energy and the energies of the first three excited states. In CHECK and THINK, comment on the difference between the third excited state and the ground ...

Verified Answer:

INTERPRET and ANTICIPATE A hockey puck in a box is...