INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH (IISER), PUNE(An Autonomous Institution, Ministry of Human Resource Development, Govt. of India)

Dr Homi Bhabha Road, Pune 411 008

FALL 2015

Assignment 5: Stationary states and particle in a 1D box potential

Course name: Quantum Mechanics 1(PHY321)

Date of submission: 14/09/2015

Stationary states are particular class of solutions of time-dependent Schrodinger equation, they have ofthe form: Ψ(x, t) = ψ(x)f(t) with f(t) = exp(−iEt/h) being a phase factor, also called the separablesolutions. Note that the spatial part of these solutions satisfies Hψ(x) = Eψ(x), the time-independentSchrodinger equation, which is an eigen-value equation. Note that, for every E value we don’t get asquare-integrable wave functions (physically acceptable) as solutions, hence leading to the quantization ofenergies. Any arbitrary solution of time dependent Schrodinger equation can be written as superposition ofthe energy eigen or stationary states (Hφn = Enφn):

ψ(x, t) =∑n

cnφn(x)e−iEnt/h. (1)

1. Particle on a ring

(a) Write down the time-independent Schrodinger equation for a particle with mass m restricted tomove on a circle of constant radius r. (1 1

2 marks)

(b) Solve the eigen value equation and obtain the stationary states (1 12 marks)

2. Particle in a boxConsider a particle in a box in a region of −a ≤ x ≤ a, is in a superposition state and is described bythe wave function

Ψ(x, 0) =1√a

[cos(πx

2a

)+ sin

(πxa

)]. (2)

(a) Write down Ψ(x, t). (1 mark)

(b) Calculate 〈x〉 and 〈x2〉 (4 marks)

3. Consider a box potential with the infinite potential barriers located at x = 0 and x = a, thenextended to infinities in both directions, and no potential in between the two points. Now use theperiodic boundary conditions for ψ(x) and its first derivative dψ(x)/dx ≡ ψ′(x): ψ(0) = ψ(a) andψ′(0) = ψ′(a) and obtain the eigen functions and eigen energies. (Note that we have used a vanishingboundary conditions in the lecture for solving 1D potential box with different box dimensions.)(2 marks)

1