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Physics · lesson 3 of 4

Schrodinger equations and a particle in a box

The time-dependent equation is iℏ ∂ψ/∂t=Hψ. A time-independent Hamiltonian permits stationary-state analysis.

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Learn the ideas

Prerequisites: Differential equations and elementary quantum states.

Evolution

The time-dependent equation is iℏ ∂ψ/∂t=Hψ. A time-independent Hamiltonian permits stationary-state analysis.

Infinite well

For a nonrelativistic particle in a one-dimensional infinite well of width L, En=n²π²ℏ²/(2mL²), n=1,2,... .

Boundaries

The infinite-well wavefunction vanishes at the walls. Its nonzero ground energy follows from boundary conditions and kinetic-energy structure.

Scaling

Energy scales inversely with mass and L² in this model. Finite wells, interactions and relativistic motion require different treatments.

Worked examples

Example 1

In a one-dimensional infinite well, find E3/E1.

Energy is proportional to n², so the ratio is 3².

9

Example 2

At fixed n and mass, width doubles. Find the new energy divided by old energy.

En is proportional to 1/L².

0.25

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 In a one-dimensional infinite well, find E3/E1.

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9

Energy is proportional to n², so the ratio is 3².

2 At fixed n and mass, width doubles. Find the new energy divided by old energy.

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0.25

En is proportional to 1/L².

3 Which n values label states in the standard one-dimensional infinite well?

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Positive integers starting at 1

The n=0 sine solution is identically zero and cannot be normalized.

4 What boundary condition holds at an infinite potential wall?

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Wavefunction vanishes at the wall

The standard infinite-well model imposes zero wavefunction at each wall.

5 If E1=2 eV, calculate E2−E1.eV

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6 eV

E2=4E1=8 eV; the gap is 6 eV.

6 Why is a stationary state not a claim that all measurable quantities are exactly known?

Sample answer and marking guidance

A stationary energy eigenstate has definite energy and time-independent probability density for a time-independent Hamiltonian, but other observables can remain uncertain.

Credit specific energy status and uncertainty in other observables.

7 At fixed n and width, mass triples. Find the new energy divided by old energy.

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1/3

En is inversely proportional to mass.

8 Give two reasons an electron in a real atom is not accurately represented by an infinite one-dimensional box.

Sample answer and marking guidance

The atomic potential is not an infinite rectangular well, and the spatial problem is three-dimensional; interactions and angular structure also matter.

Credit potential and dimensionality differences.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.