Whatever you think about José Mourinho, most would agree he knows a thing or two about football. Surveying the two epic semi-final second legs of the Champions League, he came to two very different conclusions about Liverpool’s and Tottenham’s unlikely victories.

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In the first case, he said, Liverpool’s 4-0 defeat of Barcelona at Anfield had nothing to do with strategy. Instead he attributed the remarkable turnaround – after losing 3-0 in the first leg – to “mentality”, specifically the mentality instilled in the team by Jürgen Klopp. In the second, he thought Spurs’ three second-half goals to snatch the tie against Ajax were all down to tactics, namely playing long balls up to Fernando Llorente.

Both analyses have their merits but they miss out another critical factor. Beyond all the microdramas of luck and near misses that never make it into the final result, there lies a feature of all great comeback stories: psychological momentum.

The Oxford Dictionary of Sports Science and Medicine defines psychological momentum as “the positive or negative change in cognition, affect, physiology, and behaviour caused by an event or series of events that affects either the perceptions of the competitors or, perhaps, the quality of performance and the outcome of the competition”.

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That may sound like pure psychobabble and in fact there is a lot of research that suggests psychological momentum has no notable effect on overall results. However this research, in particular Thomas Gilovich’s classic study of “streak shooting” in basketball, really only looks at how periods of hot form are just misleading clusters in the random sequences of any season. In other words, it’s not momentum but a meaningless pattern, in much the same way throwing heads four times does not change the 50-50 probability in an individual coin tossing.

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But what about psychological momentum within a single match? You can see it clearly in tennis, because the scoring system of discrete sets encourages dramatic swings in impetus. A 2012 study of a decade’s grand slams showed the most common variant of five-set matches was the one in which the winner lost the first two sets – ie, where momentum is most clearly divided.

Last week’s matches were the football equivalent of, say, the semi-final of the 2011 US Open, when Novak Djokovic lost the first two sets and saved two match points before defeating Roger Federer. Except the football versions were even more startling, because whole teams were involved and vast crowds of deafening fans. Of the two semi-finals, the most extreme momentum reversal came in the Spurs-Ajax match. To be sure, Liverpool had to beat Barcelona, with the peerless Lionel Messi, by four goals without conceding. That is a simply astonishing feat but they had several factors in their favour.

First, they played well in the first leg and were unfortunate to lose 3-0. Second, rather like a new set in tennis, the second leg was a fresh start. Third, they were playing at Anfield, where the crowd is renowned for driving on the home team. And finally, Barcelona had thrown away a three-goal lead in last year’s competition.

Facebook Twitter Pinterest Liverpool were given a huge boost in the second leg against Barcelona by their home support, seen here cheering the team bus before the game. Photograph: Tom Jenkins/The Guardian

By contrast Spurs had played poorly in the first leg, losing 1-0 at home. And to compound matters they conceded two goals in the first half in the second leg, where it was Ajax who were roared on by their home support. This year in the Champions League Spurs have made more comebacks than Frank Sinatra. They were almost out of the competition before the knockout stages – gaining just a single point in their first three matches – and were saved by a series of late goals. Then there was that extraordinary quarter-final against Manchester City, in which just about everything that could happen in a game of football did happen.

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But they had to find momentum from nowhere against Ajax, who looked more assured and astute in every department. That, paradoxically, is where the psychology reversed itself: in Ajax’s superiority. So far ahead were the Dutch they could not help but be aware the future was theirs. They had already all but won the game, so all they had to do was not lose it. And that is when they began to play as though losing was a possibility. Or if you prefer, they experienced a “negative change in cognition”.

By the same token, Spurs had nothing to lose because they were effectively already out. Yes, Llorente made a difference but it was as if clear sight of impending defeat liberated the Spurs players to start playing as if nothing else mattered. The words used by their managers to describe the Liverpool and Spurs comebacks were “belief” and “faith”. Commentators spoke about “coming back from the dead”. We live in secular times in which empirical science provides endless performance metrics and players are monitored for every last sign of physical fitness.

Yet when Lucas Moura scored the winner in the 96th minute, few people wondered what his data sets would look like. Rather it seemed like a moment of transcendence, a miracle, a morality tale of salvation or redemption played out on the hallowed turf of the Johan Cruyff Arena. No wonder Moura thanked God.

Play Video 1:45 Emotional Lucas Moura reacts to replay of his match-winning goal against Ajax – video

Of course, all of that is a trick of perception, a means of making sense of the incredible. But it was nevertheless a night that was all about belief. Tottenham did not possess it at the start, no matter what they said afterwards. They did not believe they were good enough. They did not believe they could win and they played as if they did not believe in themselves. Then they did believe. They say goals change games, and it’s true. But the momentum was already heading in a new direction before Moura scored his first goal. And it gathered pace thereafter.

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The emotions displayed by both sides at the end paid testament to the stunning change in outcome. However, for Spurs, they were the outpouring of an intense buildup of mental resolve. As a Spurs fan of half a century’s standing my psyche is decorated like a Renaissance church with scenes of torment. Yet as the clocked ticked away and the sixth minute of injury time arrived, deep in my bones I believed Spurs were going to do it. That was not fan loyalty. At half-time I was ready to turn off the TV to staunch the familiar feelings of dejection. Yet such was the feverish commitment with which Spurs fought their way back, they shifted my psychology, too. Gradually it seemed an historical inevitability that somehow they would score that third goal.

“This type of emotion,” said a tearful Mauricio Pochettino after the game, “without football is I think impossible.” He is right. Football dares you to believe and, for all the pain that comes with that, sometimes it can trigger a gloriously unstoppable momentum.