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Physics · University

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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Prerequisites

Differential equations and elementary quantum states.

Study notes

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

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

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

9

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

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

    1. Positive integers starting at 1
    2. All real numbers
    3. Only zero
    4. Only negative integers

  2. What boundary condition holds at an infinite potential wall?

    1. Wavefunction vanishes at the wall
    2. It is always constant everywhere
    3. Its magnitude must be infinite
    4. Its probability equals two

  3. If E1=2 eV, calculate E2−E1.

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

Answers and marking guidance

Answers and marking notes
  1. Positive integers starting at 1

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

  2. Wavefunction vanishes at the wall

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

  3. 6 eV

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

  4. 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.

Further reading and review references

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