Lecture 10-11 Schrödinger(薛定谔)equations

Prior to 1925 quantum physics was a “hodgepodge” of hypotheses, principles, theorems and recipes. It was not a logically consistent theory.

Once we know this wavefunction we know “everything” about the system!

Part 1Dynamic Equations

If we know the forces acting upon the particle than,according to classical physics, we know everything about a particle at any moment in the future.

A differential equation by itself does not fully determine the unknown function.

Part 2Dynamic Equation of Wave function

---- Schrödinger equations

用描述的粒子，只能有5种动量取值，分别是，对应的几率分别是，这些几率总和应该为1。

To find the expectation (average) value of p, we first need to represent p in terms of x and t. Consider the derivative of the wave function of a free particle with respect to x:

We find that

This suggests we define the momentum operator as

The expectation value of the momentum is

So,we can not have definite values for the dynamical variables, such as the momentum, when the state of a particle is determined by the wave function with respect to x. We have to find the other way to describe the dynamical variables in Quantum Mechanics.

For every dynamical variable or any observable there is a corresponding Quantum Mechanical Operator

Physical QuantityOperators

Operators are important in quantum mechanics.

All observables have corresponding operators.

OperatorsSymbols for mathematical operation

The position x is its own operator. Done. Other operators are simpler and just involve multiplication.

The potential energy operator is just multiplication by V(x).

The momentum operator is defined as

Eigenvalue equation of an operator

Deriving the Schrödinger Equation using operators:

This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree with experiment.

Schrödinger EquationNotes:

The Schrödinger Equation is THE fundamental equation of Quantum Mechanics.There are limits to its validity. In this form it applies only to a single, non-relativistic particle (i.e. one with non-zero rest mass and speed much less than c.)

a)Schrödinger equation is a linear homogeneous partial differential equation.

b)The Schrödinger equation contains the complex number i. Therefore its solutions are essentially complex (unlike classical waves, where the use of complex numbers is just a mathematical convenience.)

c)The wave equation has infinite number of solutions,some of which do not correspond to any physical or chemical reality.

  1. For electron bound to an atom/molecule, the wave function must be every where finite, and it must vanish in the boundaries
  2. Single valued
  3. Continuous
  4. Gradient (d/dr) must be continuous
  5. *d is finite, so that  can be normalized

d)Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances.

e)Conditions on the wave function:

1. In order to avoid infinite probabilities, the wave function must be finite everywhere.

2. The wave function must be single valued.

3. The wave function must be twice differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.)

4. In order to normalize a wave function, it must approach zero as x approaches infinity.

f)Only the physically measurable quantities must be real. These include the probability, momentum and energy.

Can think of the LHS of the Schrödinger equation as a differential operator that represents the energy of the particle.This operator is called the Hamiltonian of the particle, and usually given the symbol . Hamiltonian is a linear differential operator.

Hence there is an alternative (shorthand) form for the time-dependent Schrödinger equation:

Part 3 Time-independent Schrödinger equation (TISE), i.e. stationary stateSchrödinger equation

Suppose potential is independent of time

Look for a separated solution, substituteinto

•This only tells us that T(t) depends on the energy E. It doesn’t tell us what the energy actually is. For that we have to solve the space part.

•T(t) does not depend explicitly on the potential U(x). But there is an implicit dependencebecause the potential affects the possible values for the energy E.

This is the time-independent Schrödinger equation (TISE) or so-called stationary state Schrödinger equation.

Solution to full TDSE is

Even though the potential is independent of time the wavefunction still oscillates in time. But probability distribution is static

For this reason a solution of the TISE is known as a StationaryState(定态)

Stationary state Schrödinger Equation Notes:

•In one space dimension, the TISE is an ordinary differential equation (not a partial differential equation)

•The TISE is aneigenvalue equationfor the Hamiltonian operator:

Part 4 Probability current density and continuity equation

Definition of probability current density

In non-relativistic quantum mechanics, the probability currentof the wave function Ψ is defined as

in the position basis and satisfies the quantum mechanical continuity equation

with the probability density defined as

.

If one were to integrate both sides of the continuity equation with respect to volume, so that

then the divergence theorem implies the continuity equation is equivalent to the integral equation

where the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.

In particular, if is a wavefunction describing a single particle, the integral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the change of the probability of the particle being measured in V is equal to the rate at which probability flows intoV.

Derivation of continuity equation

The continuity equation is derived from the definition of probability current and the basic principles of quantum mechanics.

Suppose is the wavefunction for a single particle in the position basis (i.e. is a function of x, y, and z). Then

is the probability that a measurement of the particle's position will yield a value within V. The time derivative of this is

where the last equality follows from the product rule and the fact that the shape of V is presumed to be independent of time (i.e. the time derivative can be moved through the integral). In order to simplify this further, consider the time dependent Schrödinger equation

and use it to solve for the time derivative of :

When substituted back into the preceding equation for this gives

.

Now from the product rule for the divergence operator

and since the first and third terms cancel:

If we now recall the expression for P and note that the argument of the divergence operator is just this becomes

which is the integral form of the continuity equation.

The differential form follows from the fact that the preceding equation holds for all V, and as the integrand is a continuous function of space, it must vanish everywhere:

For all whole space we have

which means that must be continuousat any positionin the whole space.

Sothe wavefunctionand its derivative must be continuous. (An exception to this rule occurs when V is infinite.)

One more, if the andis real, the probability current over the whole 1D space which means is always continuous whatever the wavefunctionand its derivativeare continuous or not. However, has to be continuous for an acceptable physical solution for that the probability density is uniquely defined(唯一确定). As to , it may not be continuous especially at the point where the potential energy is infinite.

It is easy to prove that has to be continuous at the point where the potential energy just has a limited high step.

Have a fun!

Created by Tianxin Yang, College of Precision Instruments and Optoelectronic Engineering, TianjinUniversity Page 1 of 17