# 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.

Course: [Introduction to quantum physics and relativity](https://studybrix.com/en/courses/quantum-and-relativity/index.md)
Level: university
Prerequisites: Differential equations and elementary quantum states.

## 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.

## Practice questions

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

Answer: 9

Explanation: 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.

Answer: 0.25

Explanation: En is proportional to 1/L².

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

A. Positive integers starting at 1
B. All real numbers
C. Only zero
D. Only negative integers

Answer: Positive integers starting at 1

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

### 4. What boundary condition holds at an infinite potential wall?

A. Wavefunction vanishes at the wall
B. It is always constant everywhere
C. Its magnitude must be infinite
D. Its probability equals two

Answer: Wavefunction vanishes at the wall

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

### 5. If E1=2 eV, calculate E2−E1.

Answer: 6 eV

Explanation: 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 / 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.

Explanation: 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.

Answer: 1/3

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

Explanation: Credit potential and dimensionality differences.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.