1. Review for Exam 2 The Schrodinger Eqn.
What is important?
The Schrödinger Wave Equation
Operators, expectation values
The simple harmonic oscillator
Quantum mechanics applied to the
Hydrogen atom: quantum number, energy and angular momentum
Study Chapters 5 and 7 hard!
Erwin Schrödinger ( )

2. The Schrödinger Wave Equation
The Schrödinger Wave Equation for the wave function Y(x,t) for a particle in a potential V(x,t) in one dimension is:
where
The Schrodinger Equation is the fundamental equation of Quantum Mechanics.
Note that it’s very different from the classical wave equation.
But, except for its inherent complexity (the i), it will have similar solutions.

3. Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time: V = V(x).
The Schrödinger equation’s dependence on time and position can then be separated. Let:
And substitute into:
which yields:
Now divide by y(x) f(t):
The left side depends only on t, and the right side depends only on x. So each side must be equal to a constant. The time-dependent side is:

4. Time-Independent Schrödinger Wave Equation
Multiply both sides by f /iħ:
which is an easy differential equation to solve:
But recall our solution for the free particle:
in which f(t) = exp(-iwt), so: w = B / ħ or B = ħw, which means that: B = E !
So multiplying by y(x), the spatial Schrödinger equation becomes:

5. Stationary States The wave function can now be written as:
The probability density becomes:
The probability distribution is constant in time.
This is a standing-wave phenomenon and is called a stationary state.
Most important quantum-mechanical problems will have stationary-state solutions. Always look for them first.

6. Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is given in the equation
The probability of the particle being between x1 and x2 is given by
The wave function must also be normalized so that the probability of the particle being somewhere on the x axis is 1.

7. Expectation Values
In quantum mechanics, we’ll compute expectation values. The expectation value, , is the weighted average of a given quantity. In general, the expected value of x is:
If there are an infinite number of possibilities, and x is continuous:
Quantum-mechanically:
And the expectation of some function of x, g(x):

8. Bra-Ket Notation
This expression is so important that physicists have a special notation for it.
The entire expression is called a bracket.
And is called the bra with the ket.
The normalization condition is then:

9. General Solution of the Schrödinger Wave Equation when V = 0
In free space (with V = 0), the wave function is:
which is a sine wave moving in the x direction.
Notice that, unlike classical waves, we are not taking the real part of this function. Y is, in fact, complex.
In general, the wave function is complex.
But the physically measurable quantities must be real.
These include the probability, position, momentum, and energy.

10. Momentum Operator
To find the operator for p, consider the derivative of the wave function of a free particle with respect to x:
With k = p / ħ we have:
This yields:
This suggests we define the momentum operator as:
The expectation value of the momentum is:

11. Position and Energy Operators
The position x is its own operator. Done.
Energy operator: Note that the time derivative of the free-particle wave function is:
Substituting w = E / ħ yields:
This suggests defining the energy operator as:
The expectation value of the energy is:

12. Simple Harmonic Oscillator
Simple harmonic oscillators describe many physical situations: springs, diatomic molecules and atomic lattices.
Consider the Taylor expansion of an arbitrary potential function:
Near a minimum, V1[x-x0] ≈ 0.

13. Simple Harmonic Oscillator
Consider the second-order term of the Taylor expansion of a potential function:
Letting x0 = 0.
Substituting this into Schrödinger’s equation:
We have:
Let and , which yields:

14. The Parabolic Potential Well
The wave function solutions are: where Hn(x) are Hermite polynomials of order n.

15. The Parabolic Potential Wellyn
|yn |2
The Parabolic Potential Well
Classically, the probability of finding the mass is greatest at the ends of motion (because its speed there is the slowest) and smallest at the center.
Contrary to the classical one, the largest probability for the lowest energy states is for the particle to be at (or near) the center.
Classical result

16. Correspondence Principle for the Parabolic Potential Well
As the quantum number (and the size scale of the motion) increase, however, the solution approaches the classical result. This confirms the Correspondence Principle for the quantum-mechanical simple harmonic oscillator.
Classical result

17. The Parabolic Potential Well
The energy levels are given by:
The zero point energy is called the Heisenberg limit:

18. CHAPTER 7: Review The Hydrogen Atom
Orbital Angular momentum
Application of the Schrödinger Equation to the Hydrogen Atom
Solution of the Schrödinger Equation for Hydrogen
Spin angular momentum
Werner Heisenberg
( )
The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions. All its qualities are inferential; no material properties can be directly attributed to it. An understanding of the atomic world in that primary sensuous fashion…is impossible.
- Werner Heisenberg

19. Orbital Angular Momentum Quantum Number ℓ
Classically, angular momentum is . But quantum-mechanically, the total orbital angular momentum L is an operator, e.g.,
The eigenvalues of are functions of ℓ:
So, in an ℓ = 0 state:
This disagrees with Bohr’s semi- classical planetary model of electrons orbiting a nucleus, for which L = nħ, where n = 1, 2, …
Classical orbits—which don’t exist in quantum mechanics

20. Orbital Magnetic Quantum Number mℓ
Possible total angular momentum directions when ℓ = 2
mℓ is an integer. It determines the z component of L:
Example: ℓ = 2:
Only certain orientations of are possible.
And (except when ℓ = 0) we just don’t (and cannot!) know Lx and Ly!

21. Hydrogen-Atom Radial Wave Functions
The solutions R to the radial equation are called Associated Laguerre functions. There are infinitely many of them, for values of n = 1, 2, 3, … and ℓ < n.

22. Normalized Spherical Harmonics
Spherical harmonics are the solutions to the combination polar- and azimuthal-angle equations.

23. Complete Solution of the Radial, Angular, and Azimuthal Equations
The total wave function is the product of the radial wave function Rnℓ and the spherical harmonics Yℓm and so depends on n, ℓ, and mℓ. The wave function becomes: ℓ
where only certain values of n, ℓ, and mℓ are allowed.

24. Quantum Numbers The three quantum numbers:
n: Principal quantum number
ℓ : Orbital angular momentum quantum number
mℓ: Magnetic (azimuthal) quantum number
The restrictions for the quantum numbers:
n = 1, 2, 3, 4, . . .
ℓ = 0, 1, 2, 3, , n − 1
mℓ = − ℓ, − ℓ + 1, , 0, 1, , ℓ − 1, ℓ
n > 0
ℓ < n
|mℓ| ≤ ℓ
The energy levels are:
Equivalently:

25. Probability Distribution Functions
At the moment, we’ll consider only the radial dependence. So we should integrate over all values of q and f: The q and f integrals are just constants. So the radial probability density is P(r) = r2 |R(r)|2 and it depends only on n and ℓ.

26. Probability Distribution Functions
R(r) and P(r) for the lowest-lying states of the hydrogen atom.
Note that Rn0 is maximal at r = 0! But the r2 factor reduces the probability there to 0. Nevertheless, there’s a nonzero probability that the electron is inside the nucleus.

27. Magnetic Effects—The Zeeman Effect
In 1896, the Dutch physicist Pieter Zeeman showed that spectral lines emitted by atoms in a magnetic field split into multiple energy levels. Think of an electron as an orbiting circular current loop of I = dq / dt around the nucleus. If the period is T = 2p r / v, then: I = -e/T = -e/(2p r / v) = -e v /(2p r)
Nucleus
Image from
The current loop has a magnetic moment m = IA:
where L = mvr is the magnitude of the orbital angular momentum.
= [-e v /(2p r)] p r2 = [-e/2m] mrv

28. The Zeeman Effect
Technically, we need to go back to the Schrödinger Equation and re-solve it with this new term in the total energy.
We find that a magnetic field splits the mℓ levels. The potential energy is quantized and now also depends on the magnetic quantum number mℓ.
When a magnetic field is applied, the 2p level of atomic hydrogen is split into three different energy states with energy difference of DE = mBB Dmℓ. mℓ
Energy 1
E0 + mBB E0 −1
E0 − mBB

29. Intrinsic Spin
As with orbital angular momentum, the total spin angular momentum is:
The z-component of the spinning electron is also analogous to that of the orbiting electron:
So Sz = ms ħ
Because the magnetic spin quantum number ms has only two values, ±½, the electron’s spin is either “up” (ms = +½) or “down” (ms = -½).