Intro & PIB

Question 1

Position Operator in x direction

Answer 1

X̂

Question 2

2 non-zero wavefunctions orthogonal when

Answer 2

integral of the product of them between infinity and -infinity = 0
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Question 3

modulus of the wavefunction squared is

Answer 3

positive and real

Question 4

An observable is

Answer 4

a measureable property (e.g. bond length, dipole moment, KE)

Question 5

A Hamiltonian is

Answer 5

total energy operator (sum of kinetic energy operator T and potential energy operator V)
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Question 6

What is h bar?

Answer 6

h/2pi
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Question 7

de Broglie equation

Answer 7

λ = h/p where p=mv

Question 8

constraints of wavefunctions

Answer 8

finite
single valued
continuous
differentiable twice
wavefunction x conjugate must be integrable over all space

Question 9

Born interpretation

Answer 9

probabilistic interpretation of the wavefunction

Question 10

Born interpretation assumes

Answer 10

the wavefunction is normalised (probability of finding the particle somewhere along the x direction is 1)

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Question 11

momentum operator in x direction

Answer 11

B̂ = p̂(x) = ℏ/i d/dx

Question 12

kinetic energy operator

Answer 12

T(x) = 1/2 mv² = (mv)²/2m
sub in p̂(x)=mv
sub in p̂(x) = ℏ/i d/dx

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Question 13

general eigenvalue equation

Answer 13

B̂f = bf
(where the operator B̂ acts on the eigenfunction f to regenerate f multiplied by the eigenvalue b (a constant))

Question 14

Schrodinger equation for a free particle moving in the x direction

Answer 14

ĤΨ(x) = TΨ(x) + VΨ(x) = EΨ(x)

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Question 15

Schrodinger equation for free particle moving in x direction with zero potential energy

Answer 15

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Question 16

Wavefunction solutions to the Schrodinger equation for free particle moving in the x direction with zero potential energy and the energies associated with them

Answer 16

Ψₖ(x) = Aeⁱᵏˣ +Be⁻ⁱᵏˣ
Eₖ = k²ℏ² / 2m (operate on Ψₖ(x) with KE operator)

Question 17

Schrodinger equation for a particle between the walls moving in a one-dimensional box

Answer 17

same as for free particle as V(x) = 0

Question 18

wavefunction 0 outside the box

Answer 18

at walls the particle has infinite potential energy - impossible so the particle cannot exist outside the box so wavefunction must be 0.
Continuity of the wavefunction requires it to vanish just inside the box at x=0 and x=L (boundary conditions).

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Question 19

What are the allowed wavefunctions for a particle in a box?

Answer 19

Ψₖ(x) = Aeⁱᵏˣ + Be⁻ⁱᵏˣ
Ψₖ(x) = A(coskx + isinkx) + B(coskx - isinkx)
Ψₖ(x) = (A + B)coskx + (A - B)isinkx
C = A + B, D = (A - B)i
Ψₖ(x) = Ccoskx + Dsinkx
Ψₖ(0) = Ccos0 + Dsin0
know Ψₖ(0) = 0 so C = 0
Ψₖ(x) = Dsinkx
Ψₖ(L) = DsinkL
know Ψₖ(L) = 0 but D≠0 or Ψₖ(x) would be 0 everywhere
sinkL = 0
kL = nπ, n = 1,2,3…
therefore
Ψₙ(x) = Dsin(nπx/L), n=1,2,3…

Question 20

How can the normalisation constant D be worked out?

Answer 20

integral of Ψ²ₙ(x) = 1

Question 21

wavefunctions and energies of the particle in a box

Answer 21

Ψₙ(x) = √(2/L) sin(nπx/L) for 0≤x≤L
Eₙ = n²h² / 8mL, n=1,2,3…

Question 22

What is the normalisation constant D?

Answer 22

√(2/L)

Question 23

zero point energy

Answer 23

irremovable energy which means the particle is never stationary

Question 24

nodes that Ψₙ(x) has

Answer 24

n-1

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