Introduction to Quantum Mechanics AEP3610 Professor Scott Heinekamp

1
The Time-Independent Schrödinger Equation

• assume that the time dependence and the space
  dependence factor therefore we write Y(x,t)
  y(x)f(t) and call it separable
• everything that follows hinges on this
  assumption!!
• solutions to the TDSE can be constructed as
  linear combinations of separable solutions

• since the left side depends only on t, one can
  choose any x value and the left side is fixed
  ditto for the right side
• both sides must be a constant!! Call it E since
  its units are Joules
• and since from the derivation thats what it
  was total energy

2
Performing the separation to get two equations

• C is a constant which we will take to be 1
• the quantum state exhibits quantum oscillations
  in time at w
• the probability does not oscillate since f(t)
  f(t) C2 1
• because of this, if the TDSE separates, the
  solutions to the TISE (which we now write)
  stationary states or energy eigenstates
• linear combinations of stationary states do
  solve the TDSE, but not the TISE,and those linear
  combinations are not stationary

The right side yields the TISE

• this is the familiar TISE in the form
  operator-on-wavefunction is equal to
  energy-number-times-wavefunction

3
How to make use of solutions to the TISE

• a solution y(x) to the TISE is an eigenfunction
  of the Hamiltonian
• the associated energy is the eigenvalue
• normalization is the usual procedure assume A
  is real this will not affect the energy E
• since their time dependence is so simple, and
  their probabilities are constant in time, these
  solutions are called stationary states

• to solve TISE we need to choose a potential
  energy V(x)
• to calculate averages of physical variables we
  use their operators

• obviously, these averages are time-independent,
  so since ltxgt does not depend on t, the momentum
  average ltpgt 0
• example find the expectation value of the
  Hamiltonian!

4
What is the uncertainty in the energy, for a
stationary state?

• now to find the energy uncertainty (standard
  deviation sH)

• so, a measurement of the energy will always
  yield the value E!

What about linear combinations of stationary
states?

• we will have a whole set of solutions yE(x)
  and so a whole set of time-dependent functions
  YE(x,t) explicitly we can write

• so we can build a linear combination of the Y
  which will also solve the TDSE, but it will NOT
  solve the TISE! Why??
• with cnthe set of expansion coefficients, we
  build

5
Is such a linear combination of stationary states
itself a stationary state?

• if you know the initial state for a linear
  combo, then you know the full time dependence of
  the linear combo

• example a linear combination of two stationary
  states
• what is the time dependence, if any?

• in class show that there are times when state
  is extreme