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Multiple choice
For the 1D infinite well (
ℏ
=
m
=
1
\hbar = m = 1
ℏ
=
m
=
1
,
L
=
1
L = 1
L
=
1
), the variational trial wavefunction
ψ
=
x
(
L
−
x
)
\psi = x(L - x)
ψ
=
x
(
L
−
x
)
gives an energy
E
v
a
r
=
⟨
T
⟩
/
⟨
ψ
∣
ψ
⟩
E_{var} = \langle T\rangle/\langle\psi|\psi\rangle
E
v
a
r
=
⟨
T
⟩
/
⟨
ψ
∣
ψ
⟩
with
⟨
T
⟩
=
L
3
/
6
\langle T\rangle = L^3/6
⟨
T
⟩
=
L
3
/6
and
⟨
ψ
∣
ψ
⟩
=
L
5
/
30
\langle\psi|\psi\rangle = L^5/30
⟨
ψ
∣
ψ
⟩
=
L
5
/30
. What is
E
v
a
r
E_{var}
E
v
a
r
, and does it correctly upper-bound the exact ground state
E
1
=
π
2
/
2
≈
4.9348
E_1 = \pi^2/2 \approx 4.9348
E
1
=
π
2
/2
≈
4.9348
?
4.5 (and the bound fails)
6 (and the bound holds)
4.9348 (equal to the exact value)
5 (and yes,
5
≥
4.9348
5 \geq 4.9348
5
≥
4.9348
, the bound holds)
Check answer