Expression of type Equals

from the theory of proveit.physics.quantum.QFT

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import j, k, n
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import FourierTransform

# build up the expression from sub-expressions
expr = Equals(Qmult(NumBra(k, n), FourierTransform(n), NumKet(j, n)), Qmult(frac(one, Exp(two, frac(n, two))), Exp(e, frac(Mult(two, pi, i, j, k), Exp(two, n)))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left({_{n}}\langle k \rvert \thinspace {\mathrm {FT}}_{n} \thinspace \lvert j \rangle_{n}\right) = \left(\frac{1}{2^{\frac{n}{2}}} \thinspace \mathsf{e}^{\frac{2 \cdot \pi \cdot \mathsf{i} \cdot j \cdot k}{2^{n}}}\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9, 10
6	Literal
7	ExprTuple	11, 12
8	Operation	operator: 13 operands: 14
9	Operation	operator: 15 operand: 41
10	Operation	operator: 17 operands: 18
11	Operation	operator: 30 operands: 19
12	Operation	operator: 34 operands: 20
13	Literal
14	ExprTuple	39, 41
15	Literal
16	ExprTuple	41
17	Literal
18	ExprTuple	38, 41
19	ExprTuple	21, 22
20	ExprTuple	23, 24
21	Literal
22	Operation	operator: 34 operands: 25
23	Literal
24	Operation	operator: 30 operands: 26
25	ExprTuple	40, 27
26	ExprTuple	28, 29
27	Operation	operator: 30 operands: 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 34 operands: 35
30	Literal
31	ExprTuple	41, 40
32	Literal
33	ExprTuple	40, 36, 37, 38, 39
34	Literal
35	ExprTuple	40, 41
36	Literal
37	Literal
38	Variable
39	Variable
40	Literal
41	Variable