The Schrodinger Equation, Stationary States
and Physical Conditions



Ch 2: When things go fast, use Special
Relativity.
Ch 3: Light is a particle (photon)
Ch 4: Matter is a wave
Test 2 22OCT15 on Chapter 3 and 4.
(Note NOT Chapter 2).


Light homework this week.
Memorize the follow equations:
 (3-3)
 p. 80 (first eqn)
 (3-8)
 (4-4) and (4-5) {notice similarity to (3-2) and (3-7)}
 (4-8)
 (4-9)
 (4-10)
 (4-14) and (4-15)
 (4-20) (not the middle part)
For each equation be sure you know the meaning of
each variable and when the equation is applicable.




So far, we’ve applied Schrodinger’s equation
to a free particle.
Now we apply Schrodinger’s equation to
bound states
A bound particle is in a system with a welldefined potential energy U (x)
ℏ2 𝜕𝜕2 Ψ(𝑥𝑥,𝑡𝑡)
 −
2𝑚𝑚 𝜕𝜕𝜕𝜕 2

+ 𝑈𝑈(𝑥𝑥)Ψ(𝑥𝑥, 𝑡𝑡) =
𝜕𝜕Ψ(𝑥𝑥,𝑡𝑡)
𝑖𝑖ℏ
𝜕𝜕𝜕𝜕
Compare to free particle version:
ℏ2 𝜕𝜕 2 Ψ(𝑥𝑥, 𝑡𝑡)
𝜕𝜕Ψ(𝑥𝑥, 𝑡𝑡)
−
= 𝑖𝑖ℏ
2
2𝑚𝑚 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕


Separation of variables: Ψ 𝑥𝑥, 𝑡𝑡 = 𝜓𝜓(𝑥𝑥)𝜙𝜙(𝑡𝑡)
Insert
ℏ2 𝜕𝜕2 Ψ(𝑥𝑥,𝑡𝑡)
:−
+
2𝑚𝑚 𝜕𝜕𝜕𝜕 2
𝑈𝑈(𝑥𝑥)Ψ(𝑥𝑥, 𝑡𝑡) =
𝜕𝜕Ψ(𝑥𝑥,𝑡𝑡)
𝑖𝑖ℏ
𝜕𝜕𝜕𝜕
ℏ2
𝜕𝜕 2 𝜓𝜓(𝑥𝑥)
𝜕𝜕𝜕𝜕(𝑡𝑡)
−
𝜙𝜙(𝑡𝑡)
+ 𝑈𝑈(𝑥𝑥)𝜙𝜙(𝑡𝑡)𝜓𝜓(𝑥𝑥) = 𝑖𝑖ℏ𝜓𝜓(𝑥𝑥)
2
2𝑚𝑚
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
ℏ2
𝜕𝜕 2 𝜓𝜓(𝑥𝑥)
𝜕𝜕𝜕𝜕(𝑡𝑡)
−
𝜙𝜙(𝑡𝑡)
+ 𝑈𝑈(𝑥𝑥)𝜙𝜙(𝑡𝑡)𝜓𝜓(𝑥𝑥) = 𝑖𝑖ℏ𝜓𝜓(𝑥𝑥)
2
𝜕𝜕𝜕𝜕
2𝑚𝑚
𝜕𝜕𝜕𝜕
Divide by 𝜓𝜓(𝑥𝑥)𝜙𝜙(𝑡𝑡),
ℏ2 1 𝜕𝜕 2 𝜓𝜓(𝑥𝑥)
1 𝜕𝜕𝜕𝜕(𝑡𝑡)
−
+ 𝑈𝑈(𝑥𝑥) = 𝑖𝑖ℏ
2
𝜙𝜙(𝑡𝑡) 𝜕𝜕𝜕𝜕
2𝑚𝑚 𝜓𝜓(𝑥𝑥) 𝜕𝜕𝜕𝜕
Left is depends only on space (x), the right only on time (t)


Only possible if both sides equal the same constant C
Solution is
1 𝜕𝜕𝜕𝜕(𝑡𝑡)
𝑖𝑖ℏ
𝜙𝜙(𝑡𝑡) 𝜕𝜕𝜕𝜕
=C
𝜙𝜙 𝑡𝑡 = 𝑒𝑒 −𝑖𝑖𝜔𝜔𝑡𝑡
where 𝐶𝐶 = 𝐸𝐸 = ℏ𝜔𝜔
(memorize the solution)





Ψ 𝑥𝑥, 𝑡𝑡 = 𝜓𝜓 𝑥𝑥 𝜙𝜙 𝑡𝑡 = 𝜓𝜓 𝑥𝑥 𝑒𝑒 −𝑖𝑖𝜔𝜔𝑡𝑡
So Ψ 𝑥𝑥, 𝑡𝑡 2 = 𝜓𝜓 ∗ 𝑥𝑥 𝑒𝑒 +𝑖𝑖𝜔𝜔𝑡𝑡 𝜓𝜓 𝑥𝑥 𝑒𝑒 −𝑖𝑖𝜔𝜔𝑡𝑡 = 𝜓𝜓 𝑥𝑥
2
Does not depend on time!
Probability density (not the wave function) is
what we measure.
Particle’s probability density does not change
in time; called stationary state.

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀
ℏ2 𝜕𝜕2 𝜓𝜓 𝑥𝑥
 −
2𝑚𝑚 𝜕𝜕𝜕𝜕 2

+ 𝑈𝑈 𝑥𝑥 𝜓𝜓 𝑥𝑥 = 𝐸𝐸𝜓𝜓 𝑥𝑥
( I set C = E, and multiplied by 𝜓𝜓 𝑥𝑥 .)




Total probability of finding particle must be 1.
Ensuring this is called “normalization”.
Given by ∫𝑎𝑎𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Ψ 𝑥𝑥, 𝑡𝑡
2
𝑑𝑑𝑑𝑑 = 1
Also save function must also be smooth so we
can take two derivatives


Ch 5: 2, 3, 5, 6 and 19.
Work 19 on the board.