Physics 249 Lecture 11, Sep 28 th
2012
Reading: Chapter 6
HW3: Due Today, HW4 posted on the course homepage
1) Schrodinger equation.
For matter (massive) particles in free space 𝜓 = 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) = 𝐴𝑒 𝑖(𝑝𝑥−𝐸𝑡)/ℏ
, 𝐸 = 𝑝
2
2𝑚
, where 𝐸 = 𝜔ℏ and 𝜆 = ℎ 𝑝
= ℎ 𝑚𝑣
⇒ 𝑝 = ℎ 𝜆
= 𝑘ℏ
However, note that the relationship between E and p is such that E is first order and p is second order. This indicates the wave equation should have one derivative in time that gives energy/frequency and two in space that give momentum/wavenumber.
𝜕𝜓
𝜕𝑡
= 𝐶
𝜕 2
𝜕𝑥 𝜓
2
−𝑘 2 = −𝑖𝐶𝜔 ⇒ 𝑝 2 ℏ 2
= 𝑖𝐶
𝐸 ℏ
⇒ 𝑝 2 ℏ
= 𝑝 2
2𝑚 𝑖𝐶 ⇒ 𝐶 = −𝑖
2𝑚 ℏ
𝜕 2 𝜓
𝜕𝑥 2
= −𝑖
2𝑚 ℏ
𝜕𝜓
𝜕𝑡 or more typically write as(Schrodinger equation in free space): 𝑖ℏ
𝜕𝜓
𝜕𝑡
= − ℏ 2
2𝑚
𝜕 2
𝜕𝑥 𝜓
2
2) Schrodinger equation by quantization
The above method works but has two limitations. It doesn’t tell us how to treat a potential and it doesn’t tell us how to treat more complex situations such as dealing with energy and momentum relationships in relativity.
The solution to this problem is to recognize that the key relationship that defined the wave equation was the energy momentum relationship. In the presence of a potential this expands to including the potential energy. The second point is to recognize that given the form of a wave function you can extract the energy and momentum using partial derivatives.
𝜕𝜓
𝜕𝑡
for energy and
𝜕𝜓
𝜕𝑥
for momentum.
These ideas can be put together into a general prescription called quantization. In this case quantization means converting a classical equation into a quantum wave equation.
The prescription is as follows.

a) Start with the total energy relationship including kinetic and potential energy, which describes the physics of the system. b) Quantize the total energy relationship by making the following substitutions. 𝑝 → −𝑖ℏ
𝜕
𝜕𝑥
𝐸 → 𝑖ℏ
𝜕
𝜕𝑡
The total energy relationship becomes: 𝑝 2
𝐸 =
2𝑚
+ 𝑉 ⇒ 𝑖ℏ
𝜕𝜓
𝜕𝑡 ℏ 2
= −
2𝑚
𝜕 2 𝜓
𝜕𝑥 2
+ 𝑉𝜓
Where we now have operators p and E that can extract the momentum or energy.
This prescription can be used with any potential and either nonrelativistic or relativistic energy and momentum relationships. In fact for the photon we essentially used 𝐸 𝑐 2 𝑝 2 + 𝑚 2 𝑐 4
2
with mass zero. The photon needs the relativistic version of the total
= energy relationship since it moves at c.
3) Interpretation. Note that the form of the wave function is complex. The wave function cannot be directly measured. Instead it is used to compute probabilities such as the probability that a particle is a given location.
𝑃(𝑥, 𝑡)𝑑𝑥 = 𝜓 ∗ (𝑥, 𝑡)𝜓(𝑥, 𝑡) = |𝜓(𝑥, 𝑡)| 2 𝑑𝑥 with normalization condition
∞
∫ |𝜓(𝑥, 𝑡)|
2 𝑑𝑥 = 1
4) Potentials
In cases where there is a potential the solutions will be more complex than the solution given above for a particle in free space. We know that for instance in the case of the atom the electron must be bound to be near the atom with high probability so the solutions will be a function of r. However, we should be able to use the quantization procedure when necessary to interpret the potential.
5) Time independent Schrodinger equation.
If the potential in a problem is independent of time the wave function can be factorized into time dependent and time independent forms. 𝜓(𝑥, 𝑡) = 𝜓(𝑥)𝜙(𝑡)

𝑖ℏ
𝜕𝜓(𝑥)𝜙(𝑡)
𝜕𝑡 𝑖ℏ𝜓(𝑥)
𝜕𝜙(𝑡)
𝜕𝑡
= − ℏ 2
2𝑚
𝜕 2 𝜓(𝑥)𝜙(𝑡)
𝜕𝑥 2
+ V(x) 𝜓(𝑥)𝜙(𝑡) ℏ 2
= −
2𝑚 𝜙(𝑡)
𝜕 2 𝜓(𝑥)
𝜕𝑥 2
+ V(x) 𝜓(𝑥)𝜙(𝑡)
Dividing by the wave function:
1 𝑖ℏ 𝜙(𝑡)
𝜕𝜙(𝑡)
𝜕𝑡 ℏ 2
= −
2𝑚
1 𝜓(𝑥)
𝜕 2 𝜓(𝑥)
𝜕𝑥 2
+ V(x)
The two sides of the equation are completely independent from each other and thus must be equal to a constant.
1 𝑖ℏ 𝜙(𝑡)
𝜕𝜙(𝑡)
𝜕𝑡
= 𝐶
𝜕𝜙(𝑡) −𝑖𝐶
= 𝜙(𝑡)
𝜕𝑡 ℎ 𝜙(𝑡) = 𝑒 −𝑖𝐶𝑡/ℏ
An oscillatory function with 𝑓 = 𝐶/ℎ and since E=fh then C=E. 𝜙(𝑡) = 𝑒 −𝑖𝐸𝑡/ℏ
The time independent Schrodinger equation is: 𝑖ℏ𝜓(𝑥)
𝜕𝜙(𝑡)
𝜕𝑡 ℏ 2
𝐸𝜓(𝑥) = −
= −
2𝑚
𝜕 2 ℏ
2𝑚 𝜓(𝑥)
𝜕𝑥 2
2 𝜙(𝑡)
𝜕 2 𝜓(𝑥)
𝜕𝑥 2
+ V(x) 𝜓(𝑥)𝜙(𝑡)
+ V(x)𝜓(𝑥) = 𝐻𝜓(𝑥)
Where we have defined the Hamiltonian operator, which is related to the energy operator, and normalization condition:
∞
∫ |𝜓(𝑥)| 2 𝑑𝑥 = 1
−∞ where the wave function a) Satisfies the Schrodinger equation
(clearly!) b) 𝜓(𝑥) and
𝜕𝜓(𝑥)
are continuous, finite and single valued.
𝜕𝑥

(The wave function are being used to calculate probabilities which should very continuously from point to point at the infinitesimal level and be finite an single valued.
Also the first derivative should be continuous since the second derivative is part of the wave equation and would be infinite if there were a discontinuity) c) The wave function is small enough over all |x| that the normalization condition can be satisfied.
(Again we are calculating probabilities, which should normalize to one)
Given these conditions we will frequently violate them! For instance, it is useful to consider infinite potentials, which will introduce discontinuities in the properties of the wave function. Infinite potentials give easy to understand boundary conditions, the probability in the infinite region of the potential is zero, making it simple to solve for the wave function. They can be representative of real physical situations involving potentials large enough that a particle cannot escape the potential well. However, if we want to understand the behavior at the boundary we may have to analyze the problem more accurately.
Similarly we may be interested in time dependent systems but treat them in time independent ways. When an electron transitions between initial and final states in those two states the system are time independent, but transition itself if time dependent. If we want to understand the transition we will have to employ methods to analyze the transition more accurately.