![Page 1: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/1.jpg)
Quantum Mechanics Quantum Mechanics
![Page 2: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/2.jpg)
Wave functions and the Schrodinger equation
A stationary state has a definite energy, and can be written as
* = ||2 = “Probability distribution function”
||2 dV =
For a stationary state, • * is independent of time• * = |(x,y,z)|2
probability of finding a particle near a given point x,y,z at a time t
Particles behave like waves, so they can be described with a wave function (x,y,z,t)
![Page 3: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/3.jpg)
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80 90
100
grade
Grade distribution function
![Page 4: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/4.jpg)
The Schrodinger EquationThe Schrodinger Equation
Solving this equation will give us • the possible energy levels of a system (such as an
atom)• The probability of finding a particle in a particular
region of space
It’s hard to solve this equation. Therefore, our approach will be to learn about a few of the simpler situations and their solutions.
![Page 5: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/5.jpg)
The Schrodinger equation:The Schrodinger equation:
Kinetic energy
Potential energy
+ = Total energy
For a given U(x), • what are the possible (x)?• What are the corresponding E?
![Page 6: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/6.jpg)
For a free particle, U(x) = 0, so
(x) Aeikx
E
2k 2
2mWhere k = 2
= anything real = any value from 0 to infinity
The free particle can be found anywhere, with equal probability
Momentum: p=ħk
![Page 7: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/7.jpg)
Particle in a box
Rigid wallsNewton’s view
Potential energy function U(x)
![Page 8: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/8.jpg)
The particle in a box is not free, it is “bound” by U(x)
Examples: An electron in a long molecule or in a straight wire
“Boundary conditions”:(x) = 0 at x=0, L and all values of x outside this box, where U(x) = infinite
To be a solution of the SE, (x) has to be continuous everywhere, except where U(x) has an infinite discontinuity
![Page 9: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/9.jpg)
Solutions to the S.E. for the particle in a box
d/dx also has to be continuous everywhere, (except where U(x) has an infinite discontinuity) because you need to find d2/dx2
Normal modes of a vibrating string!
![Page 10: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/10.jpg)
From 0 < x < L, U(x) = 0, so in this region, (x) must satisfy:
2
2md2dx 2 E(x)
Same as a free particle?!?!?!?!?
![Page 11: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/11.jpg)
You may be tempted to conclude that
(x) Aeikx, the solution for a free particle, is a possible solution for the bound one too.
WRONG!!!!Why not?The above (x) does NOT satisfy the boundary conditions that (x) = 0 at x=0 and x=L.
![Page 12: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/12.jpg)
So what is the solution then?
(x) A1eikx A2e
ikx
Try the next simplest solution, a superposition of two waves
The energy again isE
2k 2
2m
![Page 13: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/13.jpg)
Rewrite (x) with sin and cos
(x) = 2iA sin(kx) = C sin(kx)
Choose values of k and that satisfy the boundary conditions:
(x) = 0 when x=0 and x=L
k = n / L = 2 / k = 2L / n
Where n = 1, 2, 3, …
![Page 14: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/14.jpg)
L =n n / 2
Each end is a node, and there can be n-1 additional nodes in between
![Page 15: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/15.jpg)
Wave functions for the particle in a box(x) = 0
![Page 16: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/16.jpg)
The energy of a particle in a box cannot be zero!
You could try to put n = 0 into this equation, but then (x) = 0, which would mean there is no particle!
![Page 17: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/17.jpg)
Wave function Probability distribution function
The function (x) = C sin(kx) is a solution to the Schrodinger Eq. for the particle in a box
![Page 18: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/18.jpg)
Q1
A. least for n = 1.
B. least for n = 2 and n = 4.
C. least for n = 5.
D. the same (and nonzero) for n= 1, 2, 3, 4, and 5.
E. zero for n = 1, 2, 3, 4, and 5.
The first five wave functions for a particle in a box are shown. The probability of finding the particle near x = L/2 is
![Page 19: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/19.jpg)
A1
A. least for n = 1.
B. least for n = 2 and n = 4.
C. least for n = 5.
D. the same (and nonzero) for n= 1, 2, 3, 4, and 5.
E. zero for n = 1, 2, 3, 4, and 5.
The first five wave functions for a particle in a box are shown. The probability of finding the particle near x = L/2 is
![Page 20: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/20.jpg)
Q2
A. least for n = 1.
B. least for n = 5.
C. the same (and nonzero) for n = 1 and n = 5.
D. zero for both n = 1 and n= 5.
Compare n=1 and n=5 states. The average value of the x-component of momentum is
![Page 21: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/21.jpg)
A2
Compare n=1 and n=5 states. The average value of the x-component of momentum is
A. least for n = 1.
B. least for n = 5.
C. the same (and nonzero) for n = 1 and n = 5.
D. zero for both n = 1 and n= 5.
The wave functions for the particle in a box are superpositions of waves propagating in opposite directions. One wave has px in one direction, the other has px in the other direction, averaging to zero.
![Page 22: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/22.jpg)
Q3
The first five wave functions for a particle in a box are shown. Compared to the n = 1 wave function, the n = 5 wave function has
A. the same kinetic energy (KE).
B. 5 times more KE.
C. 25 times more KE.
D. 125 times more KE.
E. none of the above
![Page 23: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/23.jpg)
A3
The first five wave functions for a particle in a box are shown. Compared to the n = 1 wave function, the n = 5 wave function has
A. the same kinetic energy (KE).
B. 5 times more KE.
C. 25 times more KE.
D. 125 times more KE.
E. none of the above
![Page 24: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/24.jpg)
Normalization
Not every function has this property:
If a function (x) has this property, it is “normalized”.
You can find C so that the function (x) = C sin(nx/L) is normalized.
C 2L
![Page 25: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/25.jpg)
Particle in a square well
Example: electron in a metallic sheet of thickness L, moving perpendicular to the surface of the sheet
U0 is related to the work function.
Newton: particle is trapped unless E > U0
QM: For E < U0 , the particle is “bound”
![Page 26: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/26.jpg)
(x) Acos 2mE
x Bsin 2mE
x
Inside the well (0<x<L), the solution to the SE is similar to the particle in the box (sinusoidal)
Outside, the wave function decays exponentially:(x) Cex Dex
Only for certain values of E will these functions join smoothly at the boundaries!
![Page 27: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/27.jpg)
Non-zero probability of it being outside the well! Forbidden by Newtonian mechanics.
This leads to some very odd behavior… quantum tunneling!
![Page 28: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/28.jpg)
Q4
The first three wave functions for a finite square well are shown. The probability of finding the particle at x > L is
A. least for n = 1.
B. least for n = 2.
C. least for n = 3.
D. the same (and nonzero) for n = 1, 2, and 3.
E. zero for n = 1, 2, and 3.
![Page 29: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/29.jpg)
A4
The first three wave functions for a finite square well are shown. The probability of finding the particle at x > L is
A. least for n = 1.
B. least for n = 2.
C. least for n = 3.
D. the same (and nonzero) for n = 1, 2, and 3.
E. zero for n = 1, 2, and 3.
![Page 30: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/30.jpg)
•Non-zero probability that a particle can “tunnel”through a barrier!•No concept of this from classical physics.
Potential barriers and quantum tunneling
![Page 31: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/31.jpg)
Potential barriers and quantum tunneling
• Tunnel diode in a semiconductor: Current is switched on/off ~ps by varying the height of the barrier
• Josephson junction: e- pairs in superconductors can tunnel through a barrier layer: precise voltage measurements; measure very small B fields.
• Scanning tunneling microscope (STM): view surfaces at the atomic level!
• Nuclear fusion
• Radioactive decay
Importance:
![Page 32: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/32.jpg)
Scanning tunneling microscope(~atomic force microscope)
![Page 33: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/33.jpg)
Au(100) surface : STM resolves individual atoms!
![Page 34: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/34.jpg)
A wave function for a particle tunneling through a barrier
(x) and d/dx must be continuous at 0 and L.
![Page 35: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/35.jpg)
![Page 36: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/36.jpg)
![Page 37: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/37.jpg)
Scanning tunneling microscope (STM)
Iron atoms can be arranged to make an “electron corral” (IBM’s Almaden Research Center)
![Page 38: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/38.jpg)
![Page 39: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/39.jpg)
![Page 40: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/40.jpg)
Iron on copper(111)
![Page 41: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/41.jpg)
Q5
A potential-energy function is shown. If a quantum-mechanical particle has energy E < U0, the particle has zero probability of being in the region
A. x < 0.
B. 0 < x < L.
C. x > L.
D. the particle can be found at any x
![Page 42: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/42.jpg)
A5
A. x < 0.
B. 0 < x < L.
C. x > L.
D. the particle can be found at any x
A potential-energy function is shown. If a quantum-mechanical particle has energy E < U0, the particle has zero probability of being in the region
![Page 43: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/43.jpg)
An alpha particle in a nucleus.If E > 0, it can tunnel through the barrier and escape from the nucleus.
![Page 44: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/44.jpg)
T Ge2LApprox. probability of tunneling (T<<1):
G 16 EU0
1 EU0
where
2m(U0 E)
![Page 45: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/45.jpg)
The quantum harmonic oscillator
kspring
m
U(x) 12
kspring x 2
![Page 46: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/46.jpg)
(x) Ce mkspring x 2 /2
2
2md2dx 2
12
kspring x 2 E
The Schrodinger equation for the harmonic oscillator
The solution to the SE is
Solving for E gives the energy
![Page 47: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/47.jpg)
![Page 48: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/48.jpg)
Q6
The figure shows the first six energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions
A. are nonzero outside the region allowed by Newtonian mechanics.
B. do not have a definite wavelength.
C. are all equal to zero at x = 0.
D. Both A. and B. are true.
E. All of A., B., and C. are true.
![Page 49: Quantum Mechanics · energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics](https://reader033.vdocuments.us/reader033/viewer/2022042113/5e8ea1a1938cfe69eb7773a4/html5/thumbnails/49.jpg)
A6
The figure shows the first six energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions
A. are nonzero outside the region allowed by Newtonian mechanics.
B. do not have a definite wavelength.
C. are all equal to zero at x = 0.
D. Both A. and B. are true.
E. All of A., B., and C. are true.