Penfold L2

Question 1

What is the importance of the wavefunction

Answer 1

The wavefunction completely defines the system.
If the wavefunction is known we can determine any observable property of the system.
Quantum mechanics provides the tools to determine wavefunction computationally, to interpret wavefunction and to use wavefunction to determine properties of the sytem

Question 2

How do we determine a wavefunction

Answer 2

to determine the wavefunction we define a probability of finding the particle at any point in space. We cannot define a position of the particle as in quantum mechanics a particle is distributed in space like a wave

Question 3

What is the born interpretation

Answer 3

the square of the wavefunction at any point in space is proportional to the probability of finding the particle at that point

Question 4

What is the square of the wavefunction

Answer 4

density

Question 5

Describe the two parts of a wavefunction

Answer 5

wavefunction can be complex they can have a real part and an imaginary part

Question 6

What does the real part of the wavefunction help us determine

Answer 6

amplitude

Question 7

What does the imaginary part of the wavefunction help us determine

Answer 7

the phase

Question 8

If the wavefunction at point x is 𝚿(x), what is the probability of finding the particle

Answer 8

P(x) ∝ |𝚿(x)|^2 dx

Question 9

What is |𝚿(x)|

Answer 9

the magnitude of 𝚿 at point x

Question 10

Why do we write |𝚿|^2 instead of 𝚿^2

Answer 10

because 𝚿 may be imaginary or complex so 𝚿^2 would be negative or complex but probability must be real and positive so we take the magnitude

Question 11

What is another way of writing |𝚿|^2

Answer 11

𝚿* 𝚿 where 𝚿*is the complex conjugate of 𝚿

Question 12

What do we mean when we say normalising a wavefunction

Answer 12

Normalising a wavefunction means it has been checked that the integral or the probability over all of space is 1

Question 13

Explain normalising the wavefunction

Answer 13

if you square the wavefunction and integrate over all of the region of space you are interested in, the probability of finding the particle somewhere is 1

Question 14

Describe the energy levels of quantum mechanics

Answer 14

quantum mechanics has to have discrete energy levels

Question 15

What are the restrictions on the form of the wavefuntion

Answer 15

• 𝚿 Must be continuous
• The gradient of 𝚿(d𝚿/dx) must be continuous
• 𝚿 Must have single values at any point in space
• 𝚿 must be finite everywhere
• 𝚿 cannot be zero everywhere

Question 16

What do the restrictions on 𝚿 mean

Answer 16

these restrictions on 𝚿 mean that only certain wavefunction and only certain energies of the system are allowed

Question 17

What is the uncertainty principle

Answer 17

it is impossible to specify simultaneously with precision both the momentum and the position of a particle
Δpx.Δx >/= h / 4π
Δpx - uncertainty in momentum
Δx - uncertainty in position

Question 18

How do we understand the uncertainty principle

Answer 18

The uncertainty principle imposes a fundamental limitation on how precisely we can know various observables

Question 19

What is a commutator

Answer 19

way of mathematically determining whether the order of what you are doing matters

Question 20

What does it mean if the order we do things matters

Answer 20

there is a limitation of how accurately we can measure things

Question 21

What is the commutator denoted by

Answer 21

Denoted by square brackets

Question 22

What is the operator of momentum

Answer 22

-iħ(δ/δx)

Question 23

What is the operator of position

Answer 23

x

Question 24

How do we solve the Schrödinger equation for a free particle

Answer 24

for a free particle moving in x
The potential, v=0
The particle only has kinetic energy
H𝚿 = p^2/2m 𝚿 = E 𝚿
The electron is free to have any wavelength and therefore energy, the particle is not quantised and can have any arbitrary momentum