Topic 2

Question 1

What is the time-dependent Schrödinger equation(TDSE)?

Answer 1

iℏ∂/∂tΨ(x, t) = [-ℏ/2m∂^2∂x^2 + V (x, t)]Ψ(x, t)

Question 2

What are the properties of TDSE?

Answer 2

• describes a non-relativistic, massive particle in a potential V
• is a partial differential equation, 1st order in time, 2nd order in space co-ordinates
• is linear in Ψ(x, t)

– if Ψ(x, t) is a solution of the Schördinger equation, then aΨ(x, t) is also a solution, a (some complex number)

– if Ψ1(x, t) and Ψ2(x, t) are some solutions, then aΨ1(x, t)+bΨ2(x, t) is also a solution of the Schrödinger equation, a, b – some complex numbers

Question 3

What is the wave function?

Answer 3

• Contains information about the quantum system.
• A complex function of x and t.
• By itself, Ψ does not represent any physical, observable quantity.
• a complex-valued probability amplitude
P(x, t) = |Ψ(x, t)|^2 = Ψ^∗(x, t)Ψ(x, t) = probability density of finding a particle at position x at time t
• P(x, t)dx = Ψ^∗(x, t)Ψ(x, t)dx –=probability of finding a particle between x and x + dx at
time t

Question 4

What is the general properties of the wave function?

Answer 4

Interpreting the wave function as a probability density leads to:
• total probability = 1 ⇒ Ψ(x, t) is square-integrable:
∫ ∞−∞ dxP(x, t) = ∫ ∞−∞ dx|Ψ(x, t)|^2 = 1

• Ψ(x, t) – continuous and single-valued
• Ψ’(x, t) = ∂Ψ(x,t)∂x – continuous
[except when there is an infinite discontinuity in V (x, t)]
• The behaviour |Ψ(x, t)| → 0 at large x → ±∞

Question 5

What is the time-independent Schrödinger equation (TISE)?

Answer 5

It satisfies:
[-ℏ/2m∂^2∂x^2 + V (x)]Ψ(x) = EΨ(x)

Where E - total energy, the time-independent potential is conserved

Full solution is
Ψ(x, t) = ψ(x) exp(−iEt/ℏ)