Under what conditions does quantum mechanics reduce to classical mechanics?

In summary, the conversation discusses using the equation E=(n²h²)/(8mL²) and the hint of finding the value of mL² where the change in energy between two adjacent levels is equal to kT in order to determine when quantum mechanics reduces to classical mechanics. The solution involves setting n2 = 2 and n1 = 1 and solving for mL^2, with larger values of n2 resulting in larger values of mL^2.
  • #1
amberbarton
1
0

Homework Statement


"At 310K thermal energy kT=4.28x(10^-21). Use the equation you derived above (which I worked out to be E=(n²h²)/(8mL²) )to determine under which conditions quantum mechanics reduces to classical mechanics."
The hint was that "you need to find the value of mL² for which change in E between two adjacent energy levels is equal to kT".

Homework Equations


E=kt
E=(n²h²)/(8mL²)

The Attempt at a Solution


I'm not quite sure how to start here, the only thing I've managed to do so far is:

((n2²-n1²)h²)/(8mL²)=4.28x(10^-21)

I'm not sure what to do with so many unknowns, or what the question is actually asking me to calculate!
 
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  • #2
amberbarton said:
...


I'm not quite sure how to start here, the only thing I've managed to do so far is:

((n2²-n1²)h²)/(8mL²)=4.28x(10^-21)

I'm not sure what to do with so many unknowns, or what the question is actually asking me to calculate!

Assuming the above is correct let n2 = 2 and let n1 = 1. Now solve for mL^2.

Larger values of n2 will yield larger values of mL^2 which you should point out.
 

What is quantum mechanics and how does it relate to classical mechanics?

Quantum mechanics is a branch of physics that studies the behavior of matter and energy at a very small scale, such as atoms and subatomic particles. It is a fundamental theory that describes the behavior of particles at the quantum level. Classical mechanics, on the other hand, is a branch of physics that studies the behavior of larger objects, such as planets and everyday objects. Quantum mechanics is considered the foundation of classical mechanics, as classical mechanics can be derived from quantum mechanics under certain conditions.

What are these conditions that cause quantum mechanics to reduce to classical mechanics?

The conditions that cause quantum mechanics to reduce to classical mechanics are known as the classical limit. This occurs when the object being studied is large enough that its quantum properties become negligible and can be described using classical mechanics instead. This typically happens at macroscopic scales, where the effects of quantum mechanics are too small to be observed.

Can you give an example of when quantum mechanics reduces to classical mechanics?

One example of when quantum mechanics reduces to classical mechanics is the behavior of a baseball. At a macroscopic scale, the quantum properties of the particles that make up the baseball are negligible and can be described using classical mechanics. The trajectory of the baseball can be predicted using classical mechanics equations, such as Newton's laws of motion.

What are the main differences between quantum mechanics and classical mechanics?

The main differences between quantum mechanics and classical mechanics are the scales at which they apply and the types of systems they can accurately describe. Quantum mechanics applies to objects at a very small scale, such as atoms and subatomic particles, while classical mechanics applies to larger objects. Additionally, quantum mechanics allows for particles to exist in multiple states at the same time, while classical mechanics does not.

Are there any real-world applications where quantum mechanics reduces to classical mechanics?

Yes, there are many real-world applications where quantum mechanics reduces to classical mechanics. Some examples include the behavior of everyday objects, such as cars, airplanes, and buildings. Classical mechanics is also used in engineering and technology to design and build structures and machines, as well as in the field of fluid dynamics to study the behavior of fluids. Additionally, classical mechanics is used in astronomy to describe the motion of planets and other celestial bodies.

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