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Name (Last, First):

Student ID number:

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Problem 1

Encircle all that are true.

a) Because an electron is small and has wave-like properties, its energy states

are automatically quantized. False

b) Wavefunctions can have amplitude in regions where there is no classical

probability of ﬁnding a particle. True

c) A quantum mechanical system cannot be in two eigenstates of an operator

at the same time. False

d) The operator for energy is called the Hamiltonian. True

Problem 2

An electron is trapped in a 2D square box with sides L= 1 nm. Inside the box

we have V= 0 while outside V=∞. The electron is in an eigenstate of the

Hamiltonian and the wavefunction is given as:

Ψ(x, y) = 2

Lsin 3πx

Lsin 2πy

L

a) Calculate the energy of the electron in J.

E=h2

8meL2(n2

x+n2

y) = 13h2

8meL2= 7.832 ×10−19 J.

b) How many states have the same degenerate energy?

2

c) Is there any chance of ﬁnding the electron outside of the box?

No

d) What do you expect would happen to the probability of ﬁnding the elec-

tron outside of the box if we keep the potential outside high but ﬁnite?

Now there is a ﬁnite chance of ﬁnding the electron outside of the box.

1

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