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Lecture 9: Superposition & Time-Dependent Quantum States in Quantum Mechanics - Prof. Paul, Lab Reports of Quantum Physics

University of Illinois - Urbana-Champaign Quantum Physics

Prof. Paul G. Kwiat

A part of a university lecture series on quantum mechanics. It covers the concepts of superposition and time-dependent quantum states, including the time-energy uncertainty principle and the evolution of wavefunctions. The lecture also includes examples and problems related to particle motion in a box and interference beats.

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Uploaded on 03/16/2009

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“But why must I treat the measuring device

classically? What will happen to me if I

don’t

??”

--Eugene Wigner

“There is obviously no such limitation – I can

measure the energy and look at my watch;

then I know both energy and time!”

--L. D. Landau, on the time-energy

uncertainty principle

“When I hear of Schrödinger’s cat, I reach

for my gun.” --Stephen W. Hawking

Discover Lab Reports of Quantum Physics University of Illinois - Urbana-Champaign

Partial preview of the text

Download Lecture 9: Superposition & Time-Dependent Quantum States in Quantum Mechanics - Prof. Paul and more Lab Reports Quantum Physics in PDF only on Docsity!

“But why must I treat the measuring device

classically? What will happen to me if I don’t??”

--Eugene Wigner

“There is obviously no such limitation – I can

measure the energy and look at my watch;

then I know both energy and time!”

--L. D. Landau, on the time-energy

uncertainty principle

“When I hear of Schrödinger’s cat, I reach

for my gun.” --Stephen W. Hawking

Lab 3 CommentsLab 3 Comments “Quantum Information”  One of the most modern applications of QM  quantum computing, quantum communication – cryptography, teleportation, quantum metrology  Prof. Kwiat will give an optional 214-level lecture on this topic  Sunday, March 1  3 pm, 151 Loomis

-Lab 3 meets this week. (So does Discussion, and -Lab 3 meets this week. (So does Discussion, and

there is a quiz, so don there is a quiz, so don’’t skip...)t skip...)

-If you are normally assigned to 132 Loomis, go -If you are normally assigned to 132 Loomis, go

to 257 Loomis instead. to 257 Loomis instead.

- - You will need your “Active Directory” Login

{www.ad.uiuc.edu}

-You can save a lot of time by reading the lab

ahead of time – it’s a tutorial on how to draw

wavefunctions

OverviewOverview

 Superposition of states and particle motion

 ‘Packet States’ in a Box

 Measurement in quantum physics

 Schrödinger’s Cat

 Time-Energy Uncertainty Principle

Time-Time-independentindependent SEQSEQ

 Up to now, we have considered quantum particles in

“stationary states,” and have ignored their time dependence

Remember that these special states were associated with a single energy (from solution to the SEQ)

“eigenstates”

U=∞ ψ(x) (^0) L U=∞ n= n= x n= “Functions that fit”: (λ = 2L/n) ψ(x) (^0) L x “Doesn’t fit”:

Lecture 9, Act iLecture 9, Act i We know that

What is (-i)i? a. – i b. - c. +
What is 1/i? a. – 1 b. -i c. +i
What is |e iφ | 2 ? a. 0 b. e 2iφ c. 1

Lecture 9, Act iLecture 9, Act i (-i)i = -i 2 = -(-1) = + We know that

What is (-i)i? a. – i b. - c. +
What is 1/i? a. – 1 b. -i c. +i
What is |e iφ | 2 ? a. 0 b. e 2iφ c. 1

Time-dependence of EnergyTime-dependence of Energy EigenstatesEigenstates

 Example 1: Time-evolution of an “eigenstate”

 An “eigenstate” ψ is described by a single E, so we can write: This equation has the solution: This wavefunction has a complex time-dependence, containing “i”.

But, we are mostly interested in what we measure, often |Ψ(x,t)|

As previously stated, the probability density |Ψ(x,t)| 2 associated with eigenstates of the SEQ doesn’t change with time. Thus the name for states with well-defined energies … Stationary States ψ 0 x L

Time-dependence ofTime-dependence of SuperpositionsSuperpositions with different E with different E’’ss

 It is possible that a particle can be in a superposition of

“eigenstates” with different energies.  Such superpositions are also solutions of the time-dependent SEQ!  What does it mean that a particle is “in two states”. What is its E?

Let’s see how these superpositions evolve with time.

 Consider a simple example using our trusty “particle in an

infinite square well” system:

 A particle is described by a wavefunction involving a superposition of the two lowest infinite square well states (n=1 and 2)  See nice animations at http://www.falstad.com/qm1d/ U=∞ U=∞ Ψ(x) (^0) L x

Particle Motion in a Box: ExampleParticle Motion in a Box: Example

 Consider the numerical example:

An electron in the infinite square well potential is initially (at t=0) confined to the left side of the well, and is described by the following wavefunction: If the well width is L = 0.5 nm, determine the time to it takes for the particle to “move” to the right side of the well. |ψ (x,t 0 )|^2 U=∞ U=∞ (^0) L x |ψ (x,t=0)|^2 U=∞ U=∞ (^0) L x

Particle Motion in a Box: ExampleParticle Motion in a Box: Example

 Consider the numerical example:

NormalizingNormalizing SuperpositionsSuperpositions

 It’s a mathematical fact that any two eigenstates with different eigenvalues (of any measurable, including energy) are ORTHOGONAL  Meaning:

1 2

! ( x )! ( x ) dx = 0

To normalize a superposition of normalized energy eigenstates, make the sum of the absolute squares of their coefficients equal 1. 1 2 1 2 2 2 2 2 1 2

where and have different eigenvalues then

If a b

x dx a b dx a b

|a|^2 is the probability that the particle would be found in state “ 1 ” |b|^2 is the probability that the particle would be found in state “ 2 ” |a|^2 + |b|^2 = 1 |a| 2 and |b| 2 don’t change in time because ψ 1 and ψ 2 are energy eigenstates!

Consider a particle in an infinite potential well, which at t= 0 is in the state: with ψ 2 (x) and ψ 4 (x) both normalized.

What is A 2? a. 0.5 b. 0.707 c. 0.

2. At some later time t, what is the probability density at the exact

center of the well?

a. 0 b. 1 c. It depends on the precise time t.

Lecture 9, Act 2 Lecture 9, Act 2 (^0) L x

Consider a particle in an infinite potential well, which at t= 0 is in the state: with ψ 2 (x) and ψ 4 (x) both normalized.

What is A 2? a. 0.5 b. 0.707 c. 0.

2. At some later time t, what is the probability density at the exact

center of the well?

a. 0 b. 1 c. It depends on the precise time t.

Lecture 9, Act 2 Lecture 9, Act 2 In general, the probability distribution of a superposition of energy eigenstates does depend on time. However, both of these solutions always have a node at L/2. Therefore, every possible superposition of them also has a node at L/2. As stated, the question is not well posed, since A 2 could be complex. However, let’s assume that A 2 is real (or that we were asked for |A 2 |). We are told that ψ 2 (x) and ψ 4 (x) are both normalized. Therefore:

|A 2 | 2 = 1, or |A 2 | = sqrt(1 – 0.25) = sqrt(0.75) = 0. (^0) L x

Measurements of E or xMeasurements of E or x

 The important new result concerning superpositions of

energy eigenstates is that these superpositions represent

quantum particles that are moving. Consider:

 But what happens if we try to measure E on a wavefunction

which involves more than one energy?

 We can still only measure one of the allowed energies, i.e., one of the eigenstate energies (e.g., only E 1 or E 2 in ψ(x,t) above)!

 What about measurements of |Ψ(x,t)|

2 ? If we make a large # of measurements at time t, the result should look like the probability function |Ψ(x,t)| 2 at that time. If Ψ(x,t) is normalized, |A 1 | 2 and |A 2 | 2 give us the probabilities that energies E 1 and E 2 , respectively, will be measured in an experiment! |ψ (x,t)| 2 (^0) L x

Lecture 9: Superposition & Time-Dependent Quantum States in Quantum Mechanics - Prof. Paul, Lab Reports of Quantum Physics

Related documents

Partial preview of the text

Download Lecture 9: Superposition & Time-Dependent Quantum States in Quantum Mechanics - Prof. Paul and more Lab Reports Quantum Physics in PDF only on Docsity!

“But why must I treat the measuring device

--Eugene Wigner

“There is obviously no such limitation – I can

measure the energy and look at my watch;

then I know both energy and time!”

--L. D. Landau, on the time-energy

uncertainty principle

“When I hear of Schrödinger’s cat, I reach

for my gun.” --Stephen W. Hawking

-Lab 3 meets this week. (So does Discussion, and -Lab 3 meets this week. (So does Discussion, and

there is a quiz, so don there is a quiz, so don’’t skip...)t skip...)

-If you are normally assigned to 132 Loomis, go -If you are normally assigned to 132 Loomis, go

to 257 Loomis instead. to 257 Loomis instead.

{www.ad.uiuc.edu}

-You can save a lot of time by reading the lab

ahead of time – it’s a tutorial on how to draw

wavefunctions

 Superposition of states and particle motion

 ‘Packet States’ in a Box

 Measurement in quantum physics

 Schrödinger’s Cat

 Time-Energy Uncertainty Principle

 Up to now, we have considered quantum particles in

“stationary states,” and have ignored their time dependence

“eigenstates”

 Example 1: Time-evolution of an “eigenstate”

But, we are mostly interested in what we measure, often |Ψ(x,t)|

 It is possible that a particle can be in a superposition of

Let’s see how these superpositions evolve with time.

 Consider a simple example using our trusty “particle in an

infinite square well” system:

 Consider the numerical example:

 Consider the numerical example:

NormalizingNormalizing SuperpositionsSuperpositions

! ( x )! ( x ) dx = 0

where and have different eigenvalues then

If a b

x dx a b dx a b

2. At some later time t, what is the probability density at the exact

a. 0 b. 1 c. It depends on the precise time t.

2. At some later time t, what is the probability density at the exact

a. 0 b. 1 c. It depends on the precise time t.

 The important new result concerning superpositions of

energy eigenstates is that these superpositions represent

 But what happens if we try to measure E on a wavefunction

which involves more than one energy?

 What about measurements of |Ψ(x,t)|