Cơ học lượng tử 1.2

101
Outline

The infinite square well
A comment on wavefunctions at boundaries
Parity
How to solve the Schroedinger Equation in momentum space

Please read Goswami Chapter 6.
Finite case Infinite case






Since V is not infinite in Regions 1 and 3, Since V is infinite in Regions 1 and III,
it is possible to have small damped ψ there. no ψ can exist there. ψ must terminate
abruptly at the boundaries.
102
The infinite square well

Suppose that the sides of the finite square well are extended to infinity:




It is a simplified case of the finite square well. How they differ:


103
Finite case Infinite case

To find the ψ’s use the boundary The abrupt termination of the wavefunction is conditions. nonphysical, but it is called for by this (also nonphysical) well.

Abrupt change: we cannot require dψ/dx to be continuous at boundaries. Instead, replace the boundary conditions with:

Solve for ψ in Region 2 only.
Begin with ψ = Acoskx + Bsinkx
Result: but require ψ = 0 at x = ± a/2. Result:


Comment on wavefunctions at boundaries

Anytime a wave approaches a change in potential, the wave has some probability of reflecting, regardless of whether its E is >V or
Case 1:





Case 2:





Case 3:
104
The wave will have some probability to reflect in all 3 cases.

You cannot assume there is no “Be-ikx” in Region 1 of Cases 2 and 3.

The way to see this is to work an example: Put in Aeikx + Be-ikx for Region 1 and see that the boundary conditions cannot be satisfied unless B ≠ 0.

We will have a homework about this.
105
Linear combinations of wavefunctions

Suppose that somehow a particle in a well gets into this state:

u(x)=A sin 4πx/L + B sin 6πx/L

Questions: (1) What is its Ψ(x,t)?
(2) Is this a stationary state?

Answers: (1) Recall Ψ=u(x)T(t). Here u=u1 + u2, where each ui goes with a different energy state Ei of the infinite square well:
106
107
Parity

Parity is a property of a wavefunction.
108
What determines the parity of ψ ? The form of the potential V.
The ψ’s that are eigenfunctions of some Hamiltonian H have definite parity (even or odd) if V(x)=V(-x). Prove this here:
109
So if ψ(x) is an eigenfunction of H, then so is ψ(-x).
But we said ψ(x) is non-degenerate, so ψ(-x) cannot be independent of ψ(x).
110
IV. How to solve the Schroedinger Equation in momentum space: using p-space wavefunctions.

Message: The Schroedinger Equation, and the states of matter represented by wavefunctions, are so general that theey exist outside of any particular representation (x or p) and can be treated by either.

Example: Consider a potential shaped as V(x) = Cx for x > 0
= 0 for x ≤ 0
111
112
Index by energy
113
114
How to find the allowed energies in momentum space?

Recall that any potential well produces quantized energies. We find them by applying boundary conditions (BC’s). Usually we have the BC’s expressed in x-space. So we must either
convert BC’s to p-space or
Convert A(p) to x-space.  We will do this.
115
first zero
116
Outline
What to remember from linear algebra
Eigenvalue equations
Hamiltonian operators
The connection between physics and math in Quantum Mechanics

Please re-read Goswami Section 3.3.
Please read the Formalism Supplement.
117
118
119
120
121
eigenvalue, just a number
122
123
124
125
126
SAME
127
128
129
130
131
Outline

Hilbert space
The linear algebra – Quantum Mechanics connection, continued
Dirac Notation
132
133
134
135
136
137
138
139
Outline

The Projection Operator
Position and momentum representations
Ways to understand the symbol <Φ|ψ>
Commutators and simultaneous measurements

Please read Goswami Chapter 7.
140
This is a vector.
This is a scalar.
So they can appear in any order.
Now reorder them:
141
β
Φi
c
142
143
Position representation and momentum representation

Recall that the wavefunctions exist in abstract Hilbert space. To calculate with them, we must represent them in coordinate space or momentum space.

The goal of this section: to show that “representing ψ in a space” (for example, position space) means projecting it onto each of the basis vectors of that space.

To do this we will need 2 things:
Hilbert ψ space
p-space
x-space
144
145
146
147
148
149
150
151
This means measure B first, then measure A second.
This means measure A first, then measure B second.
152
Conclusion: if we find that 2 operators have the same eigenstates, then the order of these operators’ measurements does not matter: one measurement does not disturb the system for the other. Both pieces of information can be known simultaneously.

Example of operators with simultaneous eigenstates: p and H = p2/2m + V.

We can use this information to label a state.

To uniquely label a state, list the eigenvalues of all the operators that have it as their simultaneous eigenstate.
153
Outline

The General Uncertainty Principle
The Energy-time Uncertainty Principle
The time evolution of a quantum mechanical system
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
Outline

The one-dimensional harmonic oscillator
Solving the simple harmonic oscillator using power series
171
The one-dimensional harmonic oscillator

Consider a potential of the form V = kx2/2.











Why we care about it:
it describes several real physical systems including excited nuclei, solids, and molecules. It will be generalized in field theory to describe creation and annihilation of particles.
it approximates any potential for small deviations from equilibrium. To see this, Taylor expand an arbitrary V(x) about its equilibrium point x0:
172
173
174
175
176
177
178
179
180
181
182
k is determined by the shape of the potential.

m is the mass of the bound particle.
183
Outline

Facts about the eigenfunctions and eigenvalues of the simple harmonic oscillator
More on the Hermite polynomials
Ladder operators
184
This kind of constant separation of levels can actually be observed in spectra from excited molecules. This is why those spectra are inferred to arise from vibration, that is, simple harmonic motion.
185
186
Each eigenfunction has a unique eigenvalue, that is, there is no degeneracy for this potential.

Notice this is another example of a symmetric potential producing states of definite parity.

States with n odd have odd parity.
States with n even have even parity.
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202