Mind Network - Samuel Solomon

The Schrödinger Equation

How to Use it
          The Schrödinger Equation can be very difficult to solve (for the sole reason that wave functions for molecules can be huge even after approximations). Therefore, before we even get to solving quantum systems, let us first talk about common approximations made to the Schrödinger Equation to help solve these real problems. First let us remind ourselves what the Schrödinger Equation looks like:
The Schrödinger Equation
          By just looking at the equation above, one might notice that the time derivative and the positive derivative are on different sides of the equation. We previously discussed that having all time variables on one side and all position variables on the other side is a precursor for the standing wave approximation (which uses separation of variables). In order for this to be true we must have V(x,t) = V(x). This is actually true at equilibrium (the general state of an atomic system). Nevertheless, it is not always true. Hence we will be (generally) making these two postulates:

       1. The potential is time independent (we have no changing potential field) and hence V(x,t) = V(x)
       2. The particle is a standing wave and hence psi(x,t) = p
si(x)*f(t)

These approximations can be modeled in the Schrödinger Equation below:

The Schrödinger Equation
  In order to make sure everyone is following, let us review some key steps below:
       2: The explicit approximations we will make to the Schrödinger Equation
       2 to 3: Adding the approximations from equation 2 into the Schrödinger Equation in equation 1
       3 to 4: Dividing by psi(x) * f(t). Note: the f(t) in df(t)/dt cannot be canceled as we first need to take its derivative. Notice
                    that the LHS's value cannot change if I vary time (and not position). It equals a constant

We can note that in equation 4: psi(x) (the time independent wave function) must be a twice differentiable function and continuous.

Notation note: For clarity, I wrote df(t)/dt as f '_t. All I am saying in this notation is that it is a first derivative with respect to t. I am also doing the same notation for the x derivative.

          Unit analysis will show that the constant must have units of energy. In fact, it does not look like we really changed the Schrödinger Equation at all besides the fact that the kinetic and potential energy only act on psi(x), while the total energy operator only acts on the f(t). We will therefore make another assumption:

       3. The constant in equation 4 is the total energy

We can now solve the LHS of equation 4 below:

Picture
In order to make sure everyone is following, let us review some key steps below:
       5: Rewriting the LHS of equation 4 with just the time terms and the constant
       5 to 6: We solve the differential equation
       6 to 7: E is the energy eigenvalue, which we can find through the Hamiltonian operator

          For a free particle, the energy = h_bar * omega. While this is not the energy of every particle, the time dependent term is generally seen written with it added (noting that the omega now relates to the omega of the system evaluated and not necessarily E/h_bar). Equation 7 is commonly written, but, to be familiar with all versions, we replace E/h_bar below:

Time-dependant Wave Equation
          Note: we cannot fully solve the right hand side on the equation (the position dependent differential equation) as we need to know how V depends on x. The potential matters, and we will solve these systems in the next section of the course. For now, let us leave the position dependence as its most basic form (the sum of its basis functions phi(x)).

We are also allowed to start the time (t_1) whenever we want. It is easiest to set t_1 = 0, as then t_2 is just the total time t that has occurred. This leaves us with the final equation (for some w specific to the problem's angular frequency):
Time-dependant wave equation
          The above represents the formula for a psi(x,t). All we need to do is solve the phi(x). We can note that phi(x) is time independent and hence the same at t=0 and t=t. Hence, when we solve for phi(x), m we solve for the initial condition of psi(x,t=0). Time evolving each phi(x) in the basis will yield the time dependent wave function.

As we have just recently introduced a new operator (the Hamiltonian), let us note a few facts about the operator:

       1. It is a real observable, hence the Hamiltonian is hermitian and equals its own adjoint.
       2. For a stationary state, the average energy is the eigenvalue of the Hamiltonian with 0 uncertainty.


          We can check the second point by defining what a stationary state means. A stationary state is a constant energy state with no time dependent probability distribution. This means that the probability of all observables are time independent. The uncertainty in energy is shown below:

Hamiltonian Uncertainty
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    • Introduction to Waves (The Wave Equation)
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    • Free Particle
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  • Home
  • Quantum Mechanics I
    • Introduction to Waves (The Wave Equation)
    • Introduction to Waves (The Wave Function)
    • Motivation for Quantum Mechanics (Photoelectric effect)
    • Motivation for Quantum Mechanics (Compton Scattering)
    • Motivation for Quantum Mechanics (Black Body Radiation)
    • Bohr Model of the Atom
    • Wave-Particle Duality (The Wave Function Motivation)
    • Problems with the Wave Function
    • Introduction to Quantum Operators (The Formalism)
    • Introduction to Dirac Notation
    • Introduction to Quantum Operators (The Hermitian and the Adjoint)
    • Introduction to Commutation
    • Expectation Values of Operators
    • Quantum Uncertainty (Defining Uncertainty)
    • Quantum Uncertainty (Heisenberg's Uncertainty Principle)
    • The Schrödinger Equation (The "Derivation")
    • The Schrödinger Equation (How to use it)
    • No Degeneracy in 1-Dimension
    • Parity Operator
    • Quantum Mechanics' Core Postulates
    • Free Particle (In a Vacuum)
    • Particle in a Box (Infinite Square Well)
    • Bound States (The Mathematical Setup)
    • Bound States (The Shooting Method)
    • Bound States (Patching Solutions Together)
    • Patching Solutions (Finite, Infinite, and Delta Function Potentials)
    • Delta Function Potential Well
    • Scatter States (A Touch on Dispersion)
    • Scatter States (Reflection, Transmission, Probability Current)
    • Scatter States (Worked Example)
    • Scatter States (Elastic Collision)
    • Quantum Tunneling (Constant Potential)
    • Quantum Tunneling (Changing Potential)
    • Quantum Tunneling (Alpha Decay Example)
    • Quantum Harmonic Oscillator (Classical Mechanics Analogue)
    • Quantum Harmonic Oscillator (Brute Force Solution)
    • Quantum Harmonic Oscillator (Ladder Operators)
    • Quantum Harmonic Oscillator (Expectation Values)
    • Bringing Quantum to 3D (Cartesian Coordinates)
    • Free Particle
    • Infinite Cubic Well (3D Particle in a Box)
    • Quantum Harmonic Oscillator
    • Schrödinger Equation (Spherical Coordinates)
    • Angular Momentum (Experiments)
    • Angular Momentum (Operators)
    • Angular Momentum (Ladder Operators)
    • Schrödinger Equation (Spherical Symmetric Potential)
    • Infinite Spherical Well (Radial Solution)
    • One Electron Atom (Radial Solution for S-orbital)
    • Hydrogen Atom (Angular Solution; Spherically Symmetric)
    • Hydrogen Atom (Radial Solution; Any Orbital)
    • Hydrogen atom (Recap)
  • Quantum Mechanics II
  • Nuclear Fusion
    • Introduction to Fission (Energy Extraction)
    • Introduction to Fusion (Applications and Challenges)
    • Choosing Fusion Reactants
  • Special Relativity
    • Terminology and Notation
    • Galilean Transformation
  • Statistical Thermodynamics
  • Chemical Thermodynamics
  • Ionization Radiation
  • Multivariable Calculus
    • Vectors
    • Dot Product
    • Cross Product
    • Rotating Vectors
    • Level Curves
    • Gradients
    • Directional Derivatives
  • Differential Equations
  • Contact