A marble of mass 10 g is confined to a box 10 cm long and moves at a speed of 2 cm / s. (a) W

Step 1: To find the quantum number ( n ) of the marble, we can use the formula for the energy l

A marble of mass 10 g is confined to a box 10 cm long and...

A marble of mass $10 \mathrm{~g}$ is confined to a box $10 \mathrm{~cm}$ long and moves at a speed of $2 \mathrm{~cm} / \mathrm{s}$. (a) What is the marble's quantum number $n ?$ (b) Why can we not observe the quantization of the marble's energy? [Hint: Calculate the energy difference between states $n$ and $n+1$. How much does the marble's speed change?]

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Key Concepts

Correspondence Principle

The correspondence principle states that quantum mechanics converges to classical mechanics in the limit of large quantum numbers. In the case of macroscopic objects, such as a marble in a box, the quantum effects, including discrete energy levels, become negligible, leading to behavior that appears entirely classical.

Macroscopic Limit of Quantum Effects

For systems of macroscopic scale, such as a marble, the energy differences between adjacent quantized levels become extremely small due to the large mass and size. This makes the energy spectrum effectively continuous, thereby hiding the quantum nature of the system from observable classical behavior.

Particle in a Box

The particle in a box is a fundamental quantum model where a particle is confined within a region with impenetrable boundaries. The imposition of boundary conditions requires that the particle's wavefunction vanish at the walls, resulting in quantized momentum and discrete energy levels that depend on the dimensions of the box.

Quantized Energy Levels

In quantum systems like the particle in a box, the allowed energy levels are discrete rather than continuous. These energy levels, which are proportional to the square of a quantum number, emerge from solving the Schrödinger equation with the imposed boundary conditions.

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